Optical detector device with patterned graphene layer and related methods

ABSTRACT

A method is for making an optical detector device. The method may include forming a reflector layer carried by a substrate, forming a first dielectric layer over the reflector layer, and forming a graphene layer over the first dielectric layer and having a perforated pattern therein.

RELATED APPLICATION

This application is a divisional application based on application Ser.No. 15/782,948 filed Oct. 13, 2017 which is based upon prior filedcopending Application No. 62/407,596 filed Oct. 13, 2016, the entiresubject matter of these applications is incorporated herein by referencein its entireties.

GOVERNMENT RIGHTS

This invention was made with Government support under contract No.HR0011-16-1-0003, awarded by the DARPA. The Government has certainrights in this invention.

TECHNICAL FIELD

The present disclosure relates to the field of electro-optics, and, moreparticularly, to an optical detector device and related methods.

BACKGROUND

Graphene, one of the widely studied two dimensional materials, comprisesa single layer of carbon atoms in a honeycomb lattice. It has specialelectrical, optical, and mechanical properties due to its tunable banddispersion relation and atomic thickness. Because of its unique bandstructure, graphene possesses very high mobility and fast carrierrelaxation time,¹⁻⁵ making it an attractive candidate for ultrafastelectronics and optoelectronic devices such as transistors,⁶ opticalswitches,⁷⁻⁹ mid-infrared (mid-IR) photodetectors,¹⁰ photovoltaicdevices,¹¹ saturable absorbers and ultrafast lasers¹² etc. However, lowoptical absorbance (<2.5%) in the visible to IR wavelength range makesgraphene an inefficient optical material. With such a low absorptioncross-section, these approaches are not suitable for many applications.

SUMMARY

Generally speaking, an optical detector device may include a substrate,and a reflector layer carried by the substrate. The optical detectordevice may comprise a first dielectric layer over the reflector layer,and a graphene layer over the first dielectric layer and having aperforated pattern therein.

In some embodiments, the perforated pattern may comprise a square arrayof openings. For example, each of the openings may be circle-shaped. Theperforated pattern may be symmetrical. The first dielectric layer mayhave a polymer material. The graphene layer may include a monolayer ofgraphene.

Also, the optical detector device may also include a second dielectriclayer over the graphene layer, a first electrically conductive contactcoupled to the second dielectric layer, and a second electricallyconductive contact coupled to the graphene layer. The reflector layermay comprise gold material. The reflector layer may have a thicknessgreater than a threshold thickness for optical opacity.

Another aspect is directed to a method for making an optical detectordevice. The method may include forming a reflector layer carried by asubstrate, forming a first dielectric layer over the reflector layer,and forming a graphene layer over the first dielectric layer and havinga perforated pattern therein.

BRIEF DESCRIPTION OF THE DRAWINGS

FIGS. 1a and 1b are schematic diagrams of a nanomesh device and acavity-coupled nanomesh graphene, respectively, according to the presentdisclosure.

FIGS. 1c and 1d are diagrams of a finite-difference time domain(FDTD)/coupled dipole approximation (CDA) predicted absorption and aFDTD prediction of absorption as a function of cavity thicknesses forthe cavity-coupled case, respectively, according to the presentdisclosure.

FIGS. 1e and 1f are diagrams of an optical absorption of an examplepatterned and cavity coupled patterned graphene and a FDTD predictedreal part and intensity of electric field distribution in z-direction,respectively, according to the present disclosure.

FIGS. 2a and 2b are diagrams of FDTD predicted cavity length (L) andwavelength dependent absorption, and wavelength dependent absorption atL=1.1 μm, respectively, at μ=960 cm²/V·s, according to the presentdisclosure.

FIGS. 2c and 2d are the diagrams of FDTD predicted cavity length andwavelength dependent absorption, and wavelength dependent absorption atL=1.6 μm, respectively, at μ=250 cm²/V·s, according to the presentdisclosure

FIG. 3a is a schematic diagram of a graphene plasmonic-cavity structure,according to the present disclosure.

FIG. 3b is a diagram of a Raman spectrum of grown graphene, according tothe present disclosure.

FIG. 3c is a scanning electron microscope (SEM) image of a fabricatedperforated graphene sheet on polymeric substrate, according to thepresent disclosure.

FIG. 3d is a conductive atomic-force microscopy (AFM) image of graphenenanomesh on copper foil, according to the present disclosure.

FIGS. 4a and 4b are diagrams of energy dispersion and wavelengthdependent absorption in presence of substrate phonons, and experimentaland theoretical prediction of the plasmon excitation on perforatedgraphene, respectively, according to the present disclosure.

FIGS. 5a and 5c are diagrams of tunable absorption and absorption peakshift as a function of wavelength and Fermi energy (gate voltage) forhigh mobility mono-layer graphene, according to the present disclosure.

FIGS. 5b and 5d are diagrams of tunable absorption and absorption peakshift as a function of wavelength and Fermi energy (gate voltage) forlow mobility mono-layer graphene, according to the present disclosure.

FIGS. 6a-6d are diagrams of light absorption of patterned graphene, andreal part and intensity of electric field distribution in z directionderived from FDTD for different plasmonic modes, according to thepresent disclosure.

FIGS. 7a-7b are diagrams of doping of graphene sheet by using ion gel asa dielectric for the capacitor, and an experimental result for the lightabsorption of the compound of SU-8 and ion gel, according to the presentdisclosure.

FIGS. 8a-8d are diagrams of experimental and theoretical plasmonexcitation for different Fermi energies achieved by tuning the gatevoltage, according to the present disclosure.

FIGS. 9a-9b are diagrams of electrical conductivity of graphene fordifferent types of samples, according to the present disclosure.

FIG. 10 is a diagram of light absorption of cavity coupled patternedgraphene with cavity thickness of L=1400 nm, Period=400 nm, Diameter=330nm, E_f=1.0 eV and μ=960 cm{circumflex over ( )}2/(V·s) for differentauto shutoff mins, according to the present disclosure.

FIGS. 11a-11b are diagrams of electrical conductivity of monolayergraphene sheets with different carrier mobilities, according to thepresent disclosure.

FIGS. 12a-12b are diagrams of simulated and experimental opticalabsorption spectra, respectively, according to the present disclosure.

FIG. 13 is a schematic diagram of an optical detector device, accordingto the present disclosure.

FIG. 14 is a schematic diagram of another embodiment of the opticaldetector device, according to the present disclosure.

FIG. 15 is a schematic diagram of another embodiment of the opticaldetector device, according to the present disclosure.

DETAILED DESCRIPTION

The present disclosure will now be described more fully hereinafter withreference to the accompanying drawings, in which several embodiments ofthe invention are shown. This present disclosure may, however, beembodied in many different forms and should not be construed as limitedto the embodiments set forth herein. Rather, these embodiments areprovided so that this disclosure will be thorough and complete, and willfully convey the scope of the present disclosure to those skilled in theart. Like numbers refer to like elements throughout, and base 100reference numerals are used to indicate similar elements in alternativeembodiments.

Unless graphene's absorption cross-section is dramatically enhanced,graphene will remain a scientific marvel without any practicaloptoelectronic use. An optical detector device may include a nanomeshmonolayer graphene on a dielectric layer, and a gold reflector layerunder the dielectric layer. The perforated pattern may include a squarehole array.

Although graphene's high mobility is attractive for electronic devices,the low optical absorbance along with absence of a band gap is a seriousobstacle for using graphene in optoelectronic systems. Here, Applicantsshow that it is possible to increase the light-graphene interaction andthereby enhance direct light absorption in mono-layer graphene from alow number (<2.5%) up to the unprecedented value of 60% in the midinfrared (IR) spectral domain by means of direct excitation of grapheneplasmons that are coupled to an optical cavity without using anyextraneous material. The formation of a square lattice of holes ongraphene following a simple nanoimprinting technique not only preservesmaterial continuity for electronic conductivity, which is essential foroptoelectronic devices, but also leads to direct plasmon excitation thatis independent of the incident light polarization.

Moreover, by shifting the Fermi energy and thus the density of theelectrons electrostatically, the absorption band is shown to tune over amuch wider range than previous demonstrations. Applicants developed ananalytical model that considered the effects of the electron-phononinteraction between the substrate/graphene phonons and the electrons onthe graphene, giving rise to a modified plasmon-phonon dispersionrelation which resulted in accurate correspondence between theoreticalpredictions and experimental observations. The engineered plasmon-phononinteraction decreases the edge scattering of the carriers, whichincreases the plasmon lifetime. Applicants experimentally showed thatthe enhanced absorption is minimally affected by the carrier mobilitythat is further tunable with gate voltage and cavity length. Such gatevoltage and cavity tunable enhanced absorption paves the path towardsultrasensitive infrared photodetection, optical modulation and otheroptoelectronic applications using monolayer of graphene.

Significance Statement

In this manuscript, Applicants report a direct absorption enhancementmethod based on cavity coupled patterned graphene whereby the Fermienergy is tuned by means of an external gate voltage, leading to apredicted maximum absorption of 60% and dynamic tunability up to 2 μmwhich closely corroborate experimentally measured absorption of ˜45% andtunability up to 2 μm. Such high absorption and large spectral shift inmonolayer graphene is observed, for the first time, due to the strongcoupling between localized surface plasmon resonances on the nanomeshgraphene and optical cavity modes. Such gate voltage and cavity tunableenhanced absorption paves the path towards ultrasensitive infraredphotodetection, optical modulation and other optoelectronic applicationsusing monolayer of graphene.

Introduction

Various strategies have been employed to amplify the light-matterinteraction in graphene. Excitation of surface plasmon is one suchtechnique where patterned graphene or patterned metal attached with agraphene is used to increase absorbance. In the first category ofplasmonic enhancement, graphene nanoribbons¹³⁻¹⁵ and nanodisks^(16,17)results in an enhanced absorbance of 19% and 28%, respectively. However,the discontinuity of graphene nanoribbons/disks makes these structuresimpractical for optoelectronic devices. The second approach is based onplasmonic light focusing effect where some type of metal pattern is usedto enhance the light graphene interactions^(7,8,18-21). However, withthese indirect enhancement methods only a fraction of the absorptiontakes place in the graphene, and majority of the energy is absorbed asmetal plasma loss defeating the purpose.

In contrast, Applicants employ a direct enhancement method based oncavity coupled patterned graphene whereby the Fermi energy is tuned bymeans of an external gate voltage, leading to a predicted maximumabsorption of 60% and dynamic tunability up to 2 μm which closelycoroborate experimentally measured absorption of ˜45% and gate voltagecontrolled spectral shift of ˜2 μm in monolayer graphene. Such highabsorption and large spectral shift is observed due to the strongcoupling between localized surface plasmon resonances on the nanomeshgraphene and optical cavity modes. Unlike other metal pattern basedplasmon excitations^(7,8,18-21), this direct excitation of surfaceplasmon on graphene surface ensures 100% absorption in the monolayergraphene. Moreover, absence of impurities (metals) like other indirectabsorption enhancement methods^(7,8,18-21) ensures high carriermobility.

Extraordinary Absorption Mechanism

At high EM wave frequencies in the visible domain

ω>>(E_(F),k_(B)T) where E_(F) is the Fermi energy with respect to thecharge neutrality point (CNP) of the Dirac cone, interband transitionsdominate and the light absorbance of graphene is A=πα≈2.3%, which isindependent of wavelength (α≈1/137 is the fine structure constant)⁴.However, in the mid-IR frequency range and for high Fermi energy E_(F)>>

ω, graphene's optical response is dominated by intraband transitions andthe conductivity (o) follows the Drude-Lorentz model,²⁻⁴ i.e.:

$\begin{matrix}{{\sigma^{intra}(\omega)} = \frac{{ie}^{2}\frac{E_{F}}{{\pi\hslash}^{2}}}{\omega + {i\; \tau^{- 1}}}} & (1)\end{matrix}$

where τ is the relaxation time determined by impurity scattering(τ_(imp)) and electron-phonon (τ_(el-ph)) interaction time asτ⁻¹=τ_(imp) ⁻¹+τ_(el-ph) ⁻¹ ²² (see SI).

An array of holes on graphene sheet not only conserves continuity ofgraphene, but also preserves the graphene dispersion relation andconductivity, as the edge-to-edge distance of the holes which is theshortest distance between two nearest neighbour holes is larger than themean free path of electrons. The experimentally measured carriermobilities before and after nanomesh formation was carried out tovalidate this assumption. By coupling this perforated graphene to anoptical cavity, Applicants showed that it is possible to achieveconstructive interference between incident and scattered electricfields, thereby enhancing the absorption on the graphene nanomesh.Moreover, this coupled system is able to amplify direct light absorbancein graphene even in conditions of low carrier mobility unlike othertechniques where a high carrier mobility is required for absorptionenhancement. Exciting localized surface plasmon coupled to an opticalcavity leads to strong light-matter interaction such that even in lowcarrier mobility condition the enhancement in absorption is largecompare to pristine graphene.

The system consists of a dielectric slab with variable thickness L andrefractive index n_(d) of 1.56 sandwiched between a patterned grapheneperforated with a square hole array with 330 nm diameter and 400 nmperiod and an optically thick (200 nm) gold back reflector asillustrated in FIGS. 1a-1d . These feature sizes are much larger thanthe previous 60-100 nm nanoribbon/disk patterns^(16,17) and hence mucheasier to fabricate.

A simple embossing based nanoimprinting technique was followed topattern the graphene. One such imprinting stamp can produce 1000's ofimprints without any noticeable pattern degradation. Due to thesymmetrical nanomesh square lattice pattern the excitation of LSPs isindependent of light polarization for normal angle of incidence. Thecavity thickness corresponding to a quarter wave position (L=mλ/4n_(eff)) intensifies the electric field on the graphene nanomesh dueto the constructive interference between incident and reflected fieldsinducing about two order of magnitude higher absorption in graphene.Increasing the optical cavity thickness induces higher transverse cavitymodes (L=m λ/4n_(eff)) where n_(eff) is the effective refractive indexof the dielectric slab modified by patterned graphene, which iscalculated by means of the effective medium approach^(23,24), λ is theincident EM wavelength and m=[0, 1, 2, 3, . . . ] stands for the opticalcavity m-th order. For odd/even cavity modes, the incoming and reflectedelectric fields interfere constructively/destructively at the positionof the patterned graphene, thereby giving rise to a maximum/minimumvalue in the LSP-enhanced absorbance as can be observed from the FDTDprediction in FIG. 1d for graphene with electron mobility μ=960cm²/(v·s) and Fermi energy E_(F)=1 eV excited with x-polarized light.

The corresponding FDTD predicted absorption of the patterned graphenewithout optical cavity is shown in diagram 35 FIG. 1c . The analyticalCDA^(24,25) prediction is overlaid on top of the FDTD results. The closecorrspondence between the FDTD and CDA vindicates the accuracy of theanalytical predictions (see SI for the detailed CDA derivation). Thesolid white and dotted dark lines on FDTD prediction in diagram 40 ofFIG. 1d show the analytical cavity modes dispersion as a function ofwavelength and cavity thickness which accurately demonstrates the originof this extraordinary absorption as temporal and spatial overlap betweenLSPR and cavity modes. Comparison between uncoupled and coupled systems(FIGS. 1c and 1d ) clearly demonstrates that the optical cavityintensifies the surface plasmon fields without changing the LSPresonance frequencies for all cavity modes. FIG. 1e (diagram 45)compares FDTD predicted absorption in pristine graphene, patternedgraphene and cavity-coupled patterned graphene. A cavity length of L=1.6μm which satisfies the cavity resonance condition leads to ˜60% lightabsorption around λ=10 μm, that constitutes about 30-fold absorptionenhancement compare to pristine graphene. The corresponding real part(Re (E_(z))) and intensity (|E_(z)|²) of electric field distribution inz-direction is shown in diagram 50 of FIG. if, which reveals the dipolarnature of this plasmonic mode. The side coupling in the y-directionseparates the LSP charges on the nanohole edges along the x-directionand pulls them towards the diagonal directions at 45° and −45° away fromthe x-axis. For sub-wavelength scale nearest-neighbour distance (˜λ/30)between nanoholes, coupling between them occurs via near-field andfar-field EM radiation. An optical cavity strengthens this coupling dueto the enhancement of the total electric field intensity at the edges ofnanoholes, which leads to a 5-fold increase in the optical absorptioncross-section. From field distributions of two absorption peaks (λ₀ andλ₁) in FIG. 1f , it is evident that spatial and temporal overlap betweenplasmonic and photonic cavity modes leads to strong local fieldenhancement and subsequently enhanced absorption for λ₀ (see SI fordetails).

The Effects of Carrier Mobility on the Graphene Plasmons.

Due to the two-dimensional nature of graphene, surface charge impuritiesand defects substantially alter the mean free path of electrons, andtherefore experimentally measured mobility (250-1000 cm²/(v·s)) differssignificantly from theoretically predicted range (2000-10000cm²/(v·s)).²⁶⁻²⁸ For example, polymers used to transfer the graphenesheet, the fabrication of the pattern, the doping of the graphene sheet,and oxidation decrease the electron mobility. Typically graphene on apolymer substrate has a low carrier mobility^(5,29) (<1000 cm²/(v·s))because of extra scattering processes. Typical scattering centersconsist of charge impurities, polymers roughness, and coupling betweengraphene electrons and polar or non-polar optical phonons of the polymermatrix.²⁶⁻²⁸ The reduced carrier mobility in graphene is reflected inthe reduction of the momentum relaxation time (τ), which determines theplasmon lifetime and the absorption spectrum bandwidth. FIGS. 2a-2d(diagrams 55, 60, 65, 70) show the FDTD predicted absorption spectra ofcavity-coupled nanomesh graphene for two different mobilities (p) of 960cm²/(v·s) and 250 cm²/(v·s). For relatively high mobility (960cm²/(v·s)), loss is small and therefore the bandwidth of the absorptionspectrum is narrow, indicating increased lifetime of plasmons asobserved in FIGS. 2a-2b for a cavity thickness L=1.1 μm (this cavitythickness is chosen to capture high frequency weaker resonances).

In FDTD simulation, graphene (n,k) values were obtained from mobility asdescribed in the SI. Higher loss in lower carrier mobility graphenegives rise to reduced plasmon lifetime and broadening of absorptionspectrum which results in the merger of two principal plasmonic modesand the formation of an asymmetrical peak, as seen from FIGS. 2c-2d .The first maximum absorption happens at cavity thickness L=1.6 μm forthe present geometry (FIG. 2c ). The cavity mode dispersion as afunction of cavity thickness dictates the absorption bandwidth andnumber of resonances over a specified spectral range (FIG. 2c ).Although, the light absorption decreases from 45% to 38% due to thereduction in mobility from 960 cm²/(v·s) to 250 cm²/(v·s), it is stillsubstantially larger than other strategies employed before in monolayergraphene.¹³⁻¹⁷ The carrier mobility of graphene can be adjusted byreducing the number of scattering centers on the top and the bottom ofthe graphene sheet, which further provides the possibility to tune theabsorption bandwidth and magnitude.

Nanoimprinted Cavity-Coupled Graphene.

The schematic of the cavity-coupled nanostructured graphene architectureis shown in FIG. 3a . A nanoimprinting step is performed on a spincoated polymer layer (SU-8) on top of the graphene followed by O₂ plasmaetch. Subsequent residual polymer removal steps complete the squarearray of nanohole formation on the graphene with period 400 nm anddiameter 330 nm. Due to the support from both sides, the nanoimprintingtechnique does not create cracks on the mono-layer graphene. The PDMSnanoimpriting stamp is made from a direct laser written (DLW) masterpattern. Once a master pattern is made that can produce 100's ofpolymeric imprinting stamps, and one such stamp can produce 1000's ofimprints without any noticeable pattern degradation. This method pavesthe path towards low cost production of patterned graphene. Ramanmeasurement in diagram 75 of FIG. 3b proves the presence of graphenebefore the nanomesh formation.

The Raman spectrum exhibits typical bands for single layer grapheneconsisting of a G-band at ˜1580 cm⁻¹, which is associated with doublydegenerate phonon mode (E_(2g)-symmetry) at the Γ point and originatesfrom first-order Raman scattering due to the stretching of the C—C bondin the graphene, that is prevalent in all carbon materials with sp²bands. The weak D′ peak arises from the hybridization of the G-peak,which happens when localized vibrational modes of the randomlydistributed impurities in graphene interact with its extended phononmodes. The strong 2D peak located at ˜2720 cm⁻¹ is a signature ofgraphitic sp² band materials, which is due to a second-order two-phononscattering depending on the excitation laser frequency. The shape of the2D band determines the number of graphene layers, i.e. for monolayergraphene it is sharper and more intense than the G-band in multilayergraphene³⁰⁻³². Moreover, the D-band peak commonly appears around1300-1400 cm⁻¹, which is the sign of defects and disorder in the sp²hybridized carbon structures. The D+D′ and D+D″ bands are for thesubstrate glass and polymer residue, respectively. This Ramanmeasurement in FIG. 3b confirms that the grown graphene sheet is amonolayer with some distributed impurities and defects. Diagram 80 ofFIG. 3c shows the scanning electron microscope (SEM) image of annanoimprinted nanopatterned graphene showing good uniformity in nanoholediameter across the patterned film. In order to re-confirm the nanomneshformation a conductive atomic force microscopy (AFM) was performed whichshows the holes in mono-layer graphene as a function of change inconductivity as shown in diagram 85 of FIG. 3 d.

The Effect of Plasmon-Phonon Coupling.

The simple Drude model can not capture the plasmon-phonon interactionswhich leads to a discrepancy between FDTD predictions and experimentalmeasurements. The interaction between substrate/graphene phonons andelectrons in graphene leads to modification of the graphene plasmondispersion relation, which determines the lifetime and the propagationdistance of the surface plasmon polaritons (SPPs). This coupling givesrise to novel states and band gap in the plasmonic bandstructure^(14, 33-35).

According to the random phase approximation (RPA) for a two dimensionalsystem such as graphene in quasistatic approximation, the plasmafrequency is given by^(36,37)

$\begin{matrix}{{{\omega_{p}(q)} = \sqrt{\frac{2\pi \; n_{e}e^{2}}{m^{*}ɛ_{d}}q}},} & (2)\end{matrix}$

where n_(e), e and m* are electron density, charge, and effective massin graphene, respectively. The plasmon wavevector of the nanomeshgraphene (q) is the lowest quasistatic eigenmode

${q = {\left. q_{1} \right.\sim\frac{\pi}{w - w_{0}}}},$

where w is the edge-to-edge distance of the holes and w₀ is theparameter that includes edge effects¹⁴.

By tracking the splitting of plasmon-phonon diagram and fitting thesurface plasmon-phonon polariton (SPPP) and graphene plasmon (GP)branches to the experimental data in diagram 90 of FIG. 4a , Applicantsestimated w₀=14 nm for the present w=70 nm. The white dotted curved linein FIG. 4a represents Eq. (2). A compound which consists of SU-8 ascavity spacer and ion-gel as gate dielectric layers have a longitudinaloptical phonon at ω_(sph)=36.21 THz with lifetime τ_(sph)≅2.42×10⁻¹³ s,as estimated from the absorption spectral peak location and bandwidth,respectively (FIG. 7b ). The substrate optical phonon is coupled tographene electrons by Fröhlich interaction, which leads to thehybridization of the substrate optical phonons and the grapheneplasmons. This plasmon-phonon coupling can be characterised through theloss function (Z), which is the imaginary part of inverse effectivedielectric function calculated via the generalized RPA theory^(13,14)

$\begin{matrix}{Z = {\mspace{11mu} \left( \frac{1}{ɛ_{eff}} \right)}} & (3)\end{matrix}$

The loss function represents the amount of energy dissipated by excitingthe plasmon and coupling that to the substrate and graphene opticalphonons. The details of the calculation are shown in the SI. FIG. 4ashows the loss function for graphene with mobility μ=960 cm²/(v·s) andE_(F)=1 eV. The plasmon assisted electron-hole pair generation in thisstructure is outside the Landau intraband damping region as shown as theshaded area in FIG. 4a , defined by

ω/E_(F)<q/k_(F), where k_(F) is the Fermi wavevector.³⁸ A band gap inthe plasmon-phonon dispersion relation is formed via Fröhlichinteraction between graphene plasmons and optical phonons. This couplingleads to the splitting of the energy into two distinct branches: surfaceplasmon phonon polaritons (SPPPs) and graphene phonons (GPs).^(14,15)The horizontal branch line which is marked as ω₀ is the localizedsurface plasmon mode (λ_(n)) in FIG. 1e , is independent of the plasmonwavevector. The asymmetric solid line shape of the first band in diagram95 of FIG. 4b , which is observed in experiments, is due to the mergingof these two bands.

FIG. 4a shows a clear blue shift in the GP band at a wavevector(k_(sp)=5.62×10⁵ cm⁻¹) corresponding to w=70 nm which is edge-to-edgedistance between the holes, stems from the plasmon-phonon coupling thatgives rise to the discrepancy between ω₁ and ω₂ as evident in FIG. 4b .The slight discrepancy between the theoretical prediction and theexperiment can be removed by inserting the plasmon-phonon interaction asperturbation and using ε^(rpa) in the FDTD simulation as effectivegraphene dielectric function, thereby recovering the experimentallyobserved blue shift, as illustrated in the FIG. 4b by the red dottedline. The details of this derivation are given in the SI. The thicknessof the optical cavity for the perforated graphene sheet with mobilityμ=960 cm²/(v·s) is chosen to be 1.1 μm in order to illustrate that thefirst and second modes lead to 45% and 33% light absorption,respectively. This calculation proves that the plasmon-phononinteraction hybridizes the plasmon dispersion relation which leads to ablueshift in propagating surface plasmon spectrum. However, the mainabsorption peak (ω₀) which originates from LSP remain unpurturbed due tofrequency domain seperation between phonon and LSP resonances.

Electronically Tunable Response.

FIGS. 5a and 5b (diagrams 100, 105) show theoretical and experimentalelectronically tunable absorption in cavity-coupled graphene nanomeshwith 400 nm period and 330 nm diameter for high (960 cm²/v·s) and low(250 cm²/v·s) mobilities, respectively. In both cases, Fermi energy wasvaried between 0.7 eV to 1 eV. The high and low mobility graphenesexhibit a large ˜2 μm and ˜1 μm electrostatic tunability as can beobserved in FIG. 5a and FIG. 5b , respectively. According to the Drudemodel, the refractive index of high mobility graphene is more sensitiveto change of Fermi energy than that of low mobility graphene. Thesmaller peak in FIG. 5b around 7.6 μm corresponds to polymer residue,which shows the effect of impurities in graphene optical responses.Increase in the Fermi energy leads to an increase in the electrondensity in the graphene sheet, thereby enhancing the electric dipolegenerated by LSP on the nanomesh edges. A larger electric dipole resultsin stronger coupling to the incident EM wave and therefore increase inthe light absorption irrespective of electron mobility as can be seen inFIGS. 5a-5b . Comparison between FIGS. 5a and 5b proves that the lowercarrier mobility not only leads to a decrease in the plasmon lifetime,but also causes the merging of different plasmonic modes and broaderasymmetric line-shape as predicted in FIGS. 2a-2d . The CDA model isemployed to predict the spectral shift with Fermi level. As seen fromFIGS. 5c and 5d (diagrams 110, 115), there is a good agreement betweenCDA predictions, experimental measurements, and analytical grapheneplasmon frequency ω_(p) ∝√E_(F) ∝n^(1/4 39) (detail CDA derivation is inSI).

Conclusion

Applicants have demonstrated for the first time that the directexcitation of cavity-coupled plasmon enhances the optical absorption inmono-layer graphene theoretically to around 60% and experimentallymeasured 45%, due to the strong coupling between LSP and optical cavitymodes. Applicants have shown experimentally and theoretically that thecarrier mobilty of the graphene, which is influenced by the defectdensity, determines the enhanced absorption bandwidth and line-shape.Further electronic tunability allows dynamic frequency tunable response.Such voltage tunable high absorption in mono-layer graphene will enabledevelopment of various practical graphene based optoelectronic deviceslike detectors, lasers, modulators etc.

Methods

Graphene growth: The graphene sheet is grown on a 25 μm thick copperfoil in an oven composed of a molten silica tube heated in a split tubefurnace. The molten silica tube and copper foil are loaded inside thefurnace, evacuated, back filled with hydrogen, and heated up to 1000° C.while keeping a 50 sccm H₂ stream. The subsequent steps includereinstating the copper foil at 1000° C. for 30 minutes, inserting 80sccm of CH₄ for 30 minutes. Then the furnace is cooled down to roomtemperature without gas feeding.

Cavity-couple nanoimprinted nanomesh graphene: An optically thick layerof Cr/Au (4 nm/200 nm) is deposited on a glass substrate as a backreflector using e-beam deposition. A photoresist (SU-8) layer isspin-coated on the gold back reflector to form an optical cavity, thatis cured under UV lamp for 2 hours and baked on a hot plate for 1 hourat 95° C. in order to complete the cross-linking process. A thin layer(˜20 nm) of Gold-Palladium (Au—Pd) is sputtered on the dielectric spacerwhich function as a gate electrode. A CVD-grown graphene sheet istransferred onto the Au—Pd layer using a PMMA transfer layer which issubsequently dissolved in Acetone. The square lattice hole pattern isfabricated following a simple large area nanoimprintingtechnique.^(40,41) A poly dimethylsiloxane (PDMS) stamp is embossedagainst a thin photoresist (SU-8) layer that is spun coated on thegraphene layer, followed by reactive ion etcher (RIE) in order toperforate the graphene layer. Low carrier mobility nanomesh graphene isprepared by rinsing the residual polymers (PMMA and SU-8) in acetone onetime for a few seconds. In contrast, the high carrier mobility sample isprepared by repeating this process for more than ten times in order toreduce polymer residues from the perforated graphene.

Electrostatic doping: A high capacitance ion gel film with refractiveindex of 1.3⁴² is drop-casted on graphene in order to tune its Fermienergy to high values (˜1 eV). Ion gel is a printable gate dielectricpolymer^(16,43) made by mixing ionic liquid ([EMIM][TFSI])(Sigma-Aldrich, Inc.) with dry PS-PEO-PS (10-44-10 kg/mol) triblockcopolymer (Polymer Source, Inc.) with ratio 1:0.04 in a dry solvent(dichloromethane) (Sigma-Aldrich, Inc.) and by stirring the mixtureovernight. Then it is left for 48 hours inside high vacuum chamber(pressure<10⁻⁶ torr) in order to evaporate the remaining solvent. Thematerials are dried in high vacuum for 24 hours then transferred to theglovebox for 4 days. The measured capacitance of this ion gel layer isC=1.2 μF/cm² and its absorbance in mid-IR spectrum is low. The Fermienergy of graphene is E_(F)=

v_(F)(πn)^(1/2), where v_(F)≅10⁶ m/s is the Fermi velocity and n is theelectron/hole density obtained by

${n_{e} = \frac{C\; \Delta \; V}{e}},$

where ΔV is gate voltage relative to charge neutral point (CNP). Thegate is fabricated by depositing Cr/Au (3 nm/40 nm) on Si substrate. Acopper wire is connected to the gate by applying silver paste on theside and back. The resulting substrate is flipped upside down and put ontop of the ion gel.

Conductive AFM: After RIE and the polymer removal, conductive AFM wasused to confirm the presence of a patterned graphene layer on thesubstrate. After patterning the graphene on copper foil following thesame procedure and parameters used to pattern the graphene sheet on theSU-8 layer, conductive AFM (MultiMode, Atomic Force Microscope,Nanoscope III, Digital Instruments, Santa Barbara, Calif.) is employedto map of conductivity of the patterned graphene with nanoscale spatialresolution. Conductive (Au coated) cantilevers with spring constantk=0.06 N/m was used. Measurements are performed in contact mode and afull IV curve was collected at each pixel of the image. The 1 μm*1 μmmap presented in FIGS. 3a-3d results from collecting 100*100 points.Image reconstruction was performed with Matlab.

Electromagnetic simulation: The theoretical simulations are done byfinite-difference time-domain (FDTD) method using Lumerical FDTD(Lumerical Inc.) software. The analytical coupled dipole approximation(CDA) model is developed as outlined in the SI to study the behaviour ofplasmons.

Optical measurements: The Raman spectrum of the grown graphene sheet ismeasured by WITec Renishaw RM 1000B Micro-Raman Spectrometer with anexcitation laser wavelength of 514 nm and a 50× objective lens. The realand imaginary parts of the gold dielectric function used in simulationsare taken from Palik⁴⁴. The corresponding optical absorptionmeasurements are performed with a microscope-coupled FTIR (Bruker Inc.,Hyperion 1000-Vertex 80).

A more detailed description of the drawings now follows. FIGS. 1a-1eExtraordinary absorption in cavity-coupled nanomesh graphene. FIG. 1a :Schematic of the (left) nanomesh and (right) cavity-coupled nanomeshgraphene. FIG. 1b : (left) FDTD and CDA predicted absorption of thepatterned nanomesh graphene. (right) FDTD prediction of absorption as afunction of cavity thicknesses for the cavity-coupled case. The whitesolid and green dotted lines represent constructive and destructivecavity modes, respectively. FIG. 1c : Optical absorption of pristine,patterned and cavity coupled patterned graphene with cavity thickness ofL1=1.6 μm, period=400 nm, diameter=330 nm, Ef=1.0 eV and μ=960 cm2/V·s.The first and second modes are marked by λ0 and λ1. FIG. 1d : The FDTDpredicted real part and intensity of electric field distribution inz-direction for two plasmonic modes (λ0 and λ1). The white dotted circleline shows the graphene edges.

FIGS. 2a-2d : Effect of carrier mobility on absorption enhancement. FIG.2a : FDTD predicted cavity length and wavelength dependent absorptionfor high and low mobility μ=960 cm2/V·s and FIG. 2 c: μ=250 cm2/V·s. Thecorresponding wavelength dependent absorption for two cavity thicknessesare shown in FIG. 2b and FIG. 2d respectively. Spectral broadening isobserved for the low mobility case.

FIG. 3: Fabricated system. FIG. 3a : Schematic demonstration of thegraphene plasmonic-cavity structure to excite surface plasmon fordifferent Fermi energies. FIG. 3b : The Raman spectrum of grown pristinegraphene. The presence of sharp and strong 2D peak proves mono-layergraphene. FIG. 3c : SEM image of the fabricated perforated graphenesheet on polymeric substrate (SU-8). FIG. 3d : Conductive AFM image ofgraphene nanomesh on copper foil. Change in in-plane conductivitydistinctly shows the nanomesh formation on the mono-layer graphene.

FIGS. 4a-4b : Energy loss dispersion. FIG. 4a : The energy dispersiondiagram. The colored two dimensional plot shows the evaluated lossfunction for the graphene with Ef=1.0 eV. ksp is the plasmon wavenumberassociated with the second mode. The details can be found in the SI. ω0,ω1 and ω2 represent the LSPR, the resonance propagating plasmonfrequency without and with plasmon-phonon interaction, respectively. Thecurved dotted white line is the plasmon dispersion relation of pristinesuspended graphene. The shaded area indicates the Landau intra-banddamping region. The constant pseudo-color diagram in the backgroundmarked by LSP is the loss function corresponds to the first mode whichis LSP and independent of wavenumber. FIG. 4b : The experimental andtheoretical prediction of the plasmon excitation on perforated graphenewith period=400 nm, diameter=330 nm and μ=960 cm2/V·s coupled to anoptical cavity with cavity thickness of 1.1 μm. The black solid, bluedotted and red disconnected line represent the experiment, FDTDpredicted absorption spectrum without and with plasmon-phononinteraction, respectively. The coupling of surface plasmons of grapheneand phonon polaritons of the substrate creates a new branch which ismarked as SPPP. The unperturbed branch of graphene plasmons mode is GP.The localized surface plasmon is denoted by LSP.

FIGS. 5a-5d : Dynamically tunable response. Experimentally measured andtheoretically predicted mobility dependent tunable absorption. Tunableabsorption and absorption peak shift as a function of wavelength andFermi energy (gate voltage) respectively for a (FIGS. 5a-5b ) high (960cm2/V·s) and (FIGS. 5c-5d ) low (250 cm2/V·s) mobility mono-layergraphene. A cavity thickness of 1.1 μm and 1.6 μm were chosen for highand low mobility cases respectively.

In the following, exemplary mathematic models for performance ofperforated graphene is now discussed.

Supplementary Information Calculation of Optical Extinction by CoupledDipole Approximation

For analytical calculation of the optical extinction of the perforatedgraphene in the long wavelength limit, each element is considered as anelectric dipole in the electrostatic limit with a specificpolarizability α(ω). Generally there are two different approaches toobtain α(ω) for two dimensional perforated films. The first methoddefines the polarizability of the disk element as a Lorentzian functionat the resonance frequency

$\begin{matrix}{{{\alpha (\omega)} = {\frac{3c^{3}\kappa_{r}}{2\omega_{p}^{2}}\frac{1}{\omega_{p}^{2} - \omega^{2} - \frac{i\; {\kappa\omega}^{3}}{w_{p}^{2}}}}},} & (1)\end{matrix}$

where ω_(P) is plasmon frequency of the single disk, κ is the decayrate, and κ_(r) is the radiative part of decay rate. The secondprocedure is based on the polarizability of a generalized ellipsoidalnanoparticle

$\begin{matrix}{{{\alpha (\omega)} = {ɛ_{0}V\frac{ɛ - ɛ_{m}}{ɛ + {L_{e}\left( {ɛ - ɛ_{m}} \right)}}}},} & (2)\end{matrix}$

where ε and ε_(m) are the dielectric functions of the conductive elementand surrounding medium, respectively. V defines the volume, and theshape factor of the ellipsoid, L_(e), is given by:

$\begin{matrix}{{L_{e} = {\frac{abc}{2}{\overset{\infty}{\int\limits_{0}}\frac{dq}{\left( {a^{2} + q} \right)\left\{ {\left( {q + a^{2}} \right)\left( {q + b^{2}} \right)\left( {q + c^{2}} \right)} \right\}^{\frac{1}{2}}}}}},} & (3)\end{matrix}$

where a is the diameter of the ellipsoid along the light polarizationdirection, b and c are the diameters along other two dimensions. For theperforated graphene sheet a=b=d, where d is the hole diameter and c=t,where t is the thickness of graphene.

Derivation of the LSP frequency is possible by calculation of the totalelectric potential in presence of two dimensional nanostructureelements. The relation of the induced charge (

) and the current (

) in the graphene sheet is given by the continuity equation

$\begin{matrix}{{\frac{\partial}{\partial t} + {\nabla{\cdot }}} = 0} & (4)\end{matrix}$

Due to induction of the charge density by the incoming electromagneticwave, it has exp(iωt) dependence and can be derived by means of

(r, ω)

$\begin{matrix}{{\left( {r,\omega} \right)} = {\frac{i}{\omega}{\nabla{\cdot {\left( {r,\omega} \right)}}}}} & (5)\end{matrix}$

The induced current is related to the electric potential (ϕ) by virtueof

=−σ∇ϕ, which yields the charge density

$\begin{matrix}{{\left( {r,\omega} \right)} = {{- \frac{i}{\omega}}{\nabla{\cdot {\sigma \left( {r,\omega} \right)}}}{\nabla{\varphi (r)}}}} & (6)\end{matrix}$

The total electric potential in space is due to the combination of theradiation of the graphene nanostructure and the external electric field,i.e.

$\begin{matrix}{{\varphi (r)} = {{\varphi^{ext}(r)} + {\overset{d}{\int\limits_{0}}{\overset{2\pi}{\int\limits_{0}}\frac{d^{2}r^{\prime}\left( {r^{\prime},\omega} \right)}{{r - r^{\prime}}}}}}} & (7)\end{matrix}$

Substitution of Eq. (6) in Eq. (7) gives

$\begin{matrix}{{\varphi (r)} = {{\varphi^{ext}(r)} - {\frac{i}{\omega}{\overset{d}{\int\limits_{0}}{\overset{2\pi}{\int\limits_{0}}\frac{d^{2}r^{\prime}\mspace{14mu} {\nabla^{\prime}{\cdot {\sigma \left( {r^{\prime},\omega} \right)}}}{\nabla^{\prime}{\varphi \left( r^{\prime} \right)}}}{{r - r^{\prime}}}}}}}} & (8)\end{matrix}$

By assuming homogeneous doping of graphene, its conductivity does notdepend on the position, and outside the graphene sheet the conductivitygoes to zero. It means that σ(r,ω)=

(r)σ(ω), where

(r)=1/0 for inside/outside the graphene sheet. By defining adimensionless variable

$= \frac{r}{d^{\prime}}$

the electric potential is given by

φ  ( r ) = φ ext  ( r ) +  ∫ 0 1  ∫ 0 2  π  d 2  ′   ∇ ′  · F(  - ′  ( 9 )

where

$\begin{matrix}{= {\frac{e^{2}\mspace{14mu} E_{F}}{{\pi\hslash}^{2}ɛ_{m}d}\frac{1}{\omega \left( {\omega + {i\; \tau^{- 1}}} \right)}}} & (10)\end{matrix}$

Eq. (9) introduces a self-consistent potential that in absence ofexternal potential has real eigenvalues related to plasmonic modes. TheLSP frequency is given by

$\begin{matrix}{\omega_{p} = {{\frac{e}{\hslash}\sqrt{\frac{E_{F}}{{\pi ɛ}_{m}d}}} - \frac{i}{2\tau}}} & (11)\end{matrix}$

where

is the eigenvalue of Eq. (9) and can be derived by solving thiseigensystem or by using the results from the FDTD simulation. Theimaginary part of ω_(p) is responsible for the bandwidth of theabsorption peak. In addition, eq. (11) can be applied for the graphenenanoribbon by replacing d (diameter) with w (nanoribbon width).

The lattice contribution S describes the near field and far fieldcoupling of electric dipoles

$\begin{matrix}{S = {\sum\limits_{j \neq i}\left\lbrack {\frac{\left( {1 - {ikr}_{ij}} \right)\left( {{3\cos^{2}\theta_{ij}} - 1} \right)e^{{ikr}_{ij}}}{r_{ij}^{3}} + \frac{k^{2}\sin^{2}\theta_{ij}e^{{ikr}_{ij}}}{r_{ij}}} \right\rbrack}} & (12)\end{matrix}$

where r_(ij) is the distance between electric dipoles i and j, θ_(ij) isthe angle between dipole j and {right arrow over (r_(ij))}, and kdefines the wavenumber. The optical reflection coefficient of the diskarray can be calculated by using the polarizability and the latticecontribution

$\begin{matrix}{r_{disk} = \frac{{\pm i}}{a^{- 1} - s}} & (13)\end{matrix}$

where

$\begin{matrix}{= {\frac{2\; \pi}{A}\left\{ \begin{matrix}{\left( {\cos \; \vartheta} \right)^{- 1},} & {s\text{-}{polarization}} \\{{\cos \; \vartheta},} & {p\text{-}{polarization}}\end{matrix} \right.}} & (14)\end{matrix}$

and ϑ is the incident angle, which is zero in our study, A is the areaof the unit cell, and positive/negative sign stands for s/ppolarization. The transmission coefficient of the disk array can beobtained through t_(disk)=1+r_(disk).

The absorbance (A) of the disk array on the substrate can be derived bytaking all of the reflected and transmitted electric fields at theinterface of the pattern and the substrate into account

$\begin{matrix}{A = {1 - {{r_{s} + {\left( {1 + r_{s}} \right)r_{disk}}}}^{2} - {{{Re}\left( \sqrt{\frac{ɛ_{2}}{ɛ_{1}}} \right)}{t_{s}}^{2}{{1 + r_{disk}}}^{2}}}} & (15)\end{matrix}$

where ε₁ and ε₂ are the dielectric functions of the surrounding media,and r_(s)/t_(s) denote the reflection/transmission coefficient of thesubstrate

$\begin{matrix}{r_{s} = \frac{\sqrt{ɛ_{2}} - \sqrt{ɛ_{1}}}{\sqrt{ɛ_{2}} + \sqrt{ɛ_{1}}}} & (16)\end{matrix}$

and

$\begin{matrix}{t_{s} = \frac{2\sqrt{ɛ_{1}}}{\sqrt{ɛ_{2}} + \sqrt{ɛ_{1}}}} & (17)\end{matrix}$

Substitution of the real part of Eq. (11) into Eq. (1) with

κ=12×10⁻³ eV and

κ_(r)=32.25×10⁻⁵ eV, gives the polarizability of a single disk. Thereflection coefficient of the disk array is evaluated by inserting thedisk polarizabilities in Eq. (1) and Eq. (2) into Eq. (13). Then Eq.(15) provides the two analytical absorbances. This result in diagram 120of FIG. 6a proves that the first mode corresponds to the hole-array LSP,lattice.

Analysis of the Different Plasmonic Modes

FIGS. 6a-6d are diagrams of analysis of two prominent peaks. FIG. 6a :The light absorption of patterned graphene with the carrier mobility of960 cm²/V·s, period=400 nm and diameter=330 nm without optical cavityobtained by FDTD and CDA approaches with different polarizabilities.FIG. 6b : The light absorption of cavity-coupled patterned graphene andgraphene nanoribbon with width=70 nm. The dashed blue line shows thelocation of these two peaks are same. Optical absorption of pristine,patterned and cavity coupled patterned graphene with cavity thickness ofL₁, Period=400 nm, Diameter=330 nm, E_(f)=1.0 eV and μ=960 cm²/V·s. Thefirst, second and third modes are shown by λ₁, λ₂ and λ₃, respectively.The inset image shows different regions which are responsible for thosemodes. FIG. 6d : The real part and intensity of electric fielddistribution in z direction derived from FDTD for different plasmonicmodes. The white circle line shows the graphene edges.

According to FDTD results, the plasmon frequency of a graphenenanoribbon array with period=400 nm and width=70 nm, which is equal tothe edge-edge distance of the holes, is equal to the resonance frequencyof the third mode, as seen from diagram 125 of FIG. 6b . FIG. 6c(diagram 130) demonstrates three prominent absorption peaks appear dueto exciting surface plasmons. The first mode labeled by λ₁. The redcolored area has a much larger average width than the green coloredarea, and thus in accordance with Eq. (11) the plasmon resonancewavelength corresponding to the red colored area should be longer.Moreover, this mode is similar in shape with the mode of a squarelattice of graphene nanodisks. The second mode (λ₂) coincides with thegreen region, which looks like the surface plasmon mode of a graphenenanoribbon. The plasmon resonance wavelength of an array of graphenenanoribbons with same width (70 nm) and period (400 nm) corresponds tothe second peak. The nanoribbon array has a larger effective crosssection to interact with the EM wave than the green region of theperforated sheet. That is why to the nanoribbon has a greater absorbanceof 85%. The third mode (λ₃) exhibits an electric field distribution thatchanges sign back and forth in y direction, as shown in diagram 135 ofFIG. 6d . The dipole strength of the mode λ₁ is substantially larger(i.e. >10%) than the modes λ₂ and λ₃ due to the larger cross section,which provides a higher electron density for the absorbance. Accordingto FIGS. 1c-1d of the main manuscript, different modes can beintensified by choosing the appropriate cavity thickness, which providesanother degree of tunability for this architecture. The different modesat lower wavelengths emerge because of diffraction of surface EM waves.For graphene in an asymmetric dielectric medium, the plasmon wavenumber(k_(p)) can be calculated by means of

$\begin{matrix}{{\frac{ɛ_{1}}{q_{z\; 1}} + \frac{ɛ_{2}}{q_{z\; 2}} + {2{\sigma^{intra}(\omega)}}} = 0} & (18)\end{matrix}$

where ε₁ and ε₂ are dielectric functions of adjacent environments,q_(z1,2)=√{square root over (ε_(1,2)−(k_(p)/k)²)} and k is thewavenumber of incident EM wave. The diffraction orders correspond to thesolutions of Eq. (18) which leads to appearing different peaks at lowerwavelengths.

Plasmonic structures can be used to enhance the spontaneous emissionrate due to wavelength confinement and amplification of the light-matterinteraction. The enhancement of the spontaneous emission rate isdetermined by Q/V_(eff) where Q is the quality factor given by the ratioof resonance frequency and peak bandwidth (ω_(p)/Δω). The mode volume,derived via the EM field distribution, divided by the free space modevolume (λ₀ ³) is equal to the effective mode volume V_(eff). Thecalculated spontaneous emission enhancement for various modes and Fermienergies ranges from 10⁷ to 10⁸, which constitutes a 3 orders ofmagnitude increase relative to the simple metal plasmonic structureowing to the atomic thickness, the small loss of graphene, and theoptical cavity.

Absorption of Substrate and Superstrate

FIG. 7a is a diagram of doping of graphene sheet by using ion gel as adielectric for the capacitor. Charge neutrality point (CNP) is shown inthe image. In FIG. 7b , the experimental result for the light absorptionof the compound of SU-8 and ion gel.

The ion gel is used as dielectric to fabricate a capacitor for dopinggraphene electrostatically. The absorption of the compound of ion geland SU-8 is shown in FIG. 7b (diagram 145), and the resistance is shownin diagram 140 of FIG. 7a . This compound has an absorption peak at thefrequency ω≈36.2 THz with bandwidth Δω≈10 THz, which is due to theexcitation of longitudinal optical phonons. At higher wavelengths theabsorption is around 50%, which is saturated for graphene absorbancelarger than 50%. To obtain the pure light absorption by nanomeshgraphene, the total absorption is normalized by the absorption of thiscompound.

FIGS. 9a-9b are diagrams of electrical conductivity of graphene fordifferent types of samples. Comparison of experimental results andanalytical equations indicates μ=250 cm²/V·s and μ=960 cm²/V·s,respectively and the amount of Fermi energies correspond to differentapplied gate voltages.

The measured capacitance of the ion gel layer is C=2.4 μF/cm² and itsabsorption in mid-IR spectrum is low. The Fermi energy of graphene isE_(F)=

v_(F)(πn)^(1/2), where v_(F)≅10⁶ m/s is the Fermi velocity and n is theelectron/hole density obtained by n_(e)=CΔV/e, where ΔV is gate voltagerelative to charge neutral point (CNP). The reported Fermi eneries arecalculated based on this relation. To prove the corresponding Fermienergies experimentally, the conductivity of graphene sheet is calcualedbased on σ(E_(F))=σ_(min)(1+E_(F) ⁴/Δ⁴)^(1/2),⁴⁵ where σ_(min) is theminimum conductvity and A is the disorder strength parameter. As shownin diagrams 170, 175 of FIGS. 9a-9b , by fitting this conductivity tothe experimental data (red dotted line), σ_(min)=0.289/0.371 ms andΔ=297 meV/177 meV for FIGS. 9a-9b , respectively. The relation betweenconductivity and mobility is σ=neμ, where μ is the carrier mobility ofgraphene. Fitting this equation (green solid line) to the experimentalresults in FIGS. 9a-9b illustrates μ=250/960 cm²/V·s for FIGS. 9a-9b ,respectively. Positive and negative gate voltages correspond to n-dopedand p-doped graphene, with a minimum conductivity occurring at thecharge neutral point (CNP). According to these diagrams, the graphenesheet is doped a little bit during growth and transfer (0.05 eV).

Electrostatic Tuning of Absorption

FIGS. 8a-8d are diagrams of experimental and theoretical plasmonexcitation for different Fermi energies achieved by tuning the gatevoltage. FIG. 8a : The first plasmonic mode of the cavity-coupledpatterned graphene with the carrier mobility of 960 cm²/V·s, period=400nm, diameter=330 nm and cavity thickness=1.1 μm. The green dotted lineshows the electronic tunability of the plasmon peaks. FIG. 8b : Theelectronic tuning of cavity-coupled patterned graphene with the carriermobility of 250 cm²/V·s, period=400 nm, diameter=330 nm and cavitythickness=1.6 μm. FIGS. 8c-8d : The electronic tuning of cavity-coupledpatterned graphene with the carrier mobility of 250 cm²/V·s, period=310nm, diameter=200 nm and cavity thickness=1.6 μm.

According to Eq. (11), increasing the Fermi energy leads to blue shiftof the resonance frequency, which are related to the graphene nanomeshwith mobility μ=960 cm²/(V·s) and μ=250 cm²/(V·s), respectively. Anothergraphene sample with 250 cm²/(V·s) mobility is perforated with 310 nmperiod and 200 nm diameter. This structure excites the LSP at thefrequencies larger than that of the sample with 400 nm period and 330 nmdiameter, which demonstrates another way for tuning the resonancefrequency. Moreover, the plasmon frequency of this new structure istunable by changing the Fermi energy. The constant peaks at lowerwavelengths confirm the presence of the polymer residual. These resultsare shown in diagrams 150, 155, 160, 165 of FIGS. 8a -8 d.

Calculating Loss Function and Effective Refractive Index of Graphene inPresence of Substrate

In the random-phase approximation (RPA), for high frequencies thecomplex graphene conductivity is given by

$\begin{matrix}{{\sigma (\omega)} = {\frac{e^{2}\omega}{i\; \pi \; \hslash}\left\lbrack {{\int_{- \infty}^{+ \infty}\ {d\; ɛ\frac{ɛ}{\omega \left( {\omega + {i\; \tau^{- 1}}} \right)}\frac{d\; {n_{F}(ɛ)}}{d\; ɛ}}} - {\int_{- \infty}^{+ \infty}{d\; ɛ\frac{{n_{F}\left( {- ɛ} \right)} - {n_{F}(ɛ)}}{\left( {\omega + {i\; \delta}} \right)^{2} - {4\; ɛ^{2}}}}}} \right\rbrack}} & (19)\end{matrix}$

where δ→0 is the infinitesimal parameter that is used to bypass thepoles of the integral. The first and second terms correspond to theintraband electron-photon scattering processes and direct electroninterband transitions, respectively. By taking the first integral, theintraband scattering is similar to the Drude conductivity

σ intra  ( ω ) = i  2  e 2  k B  T π   2  ( ω + i   τ - 1 ) ln  ( 2  cosh  E F 2  k B  T ) ( 20 )

where k_(B) is the Boltzmann constant and T is the temperature. At lowtemperatures k_(B)T<<E_(F), the graphene conductivity follows the Drudemodel

σ intra  ( ω ) ≈ i  e 2  E F π  2  ( ω + i   τ - 1 ) ( 21 )

According to the charge conservation law, the relation of the bulkcurrent J_(V) and the surface current J_(S) for a material is given by

∫∫J _(S) ds=∫∫∫J _(V) dV  (22)

which means the relation of two and three dimensional conductivity isdefined by

$\begin{matrix}{\sigma_{3D} = \frac{\sigma_{2D}}{t}} & (23)\end{matrix}$

where t describes the thickness of the material. The dielectric functionof graphene can be obtained via its AC conductivity by means of

$\begin{matrix}{{ɛ(\omega)} = {ɛ_{g} + \frac{i\; \sigma_{3\; D}}{ɛ_{0}\omega}}} & (24)\end{matrix}$

where ε_(g)=2.5 is the dielectric constant of graphite.

Substituting Eq. (23) into Eq. (24) gives the in-plane dielectricfunction of graphene, i.e.

$\begin{matrix}{{ɛ(\omega)} = {{ɛ_{g} + \frac{i\; \sigma^{intra}}{ɛ_{0}\omega \; t}} = {ɛ_{g} - \frac{e^{2}E_{F}}{\pi \; {\;}^{2}ɛ_{0}{\omega \left( {\omega + {i\; \tau^{- 1}}} \right)}t}}}} & (25)\end{matrix}$

whereas the surface-normal component is ε_(z)=2.5.

The dynamical polarization

$\begin{matrix}{{P\left( {q,{i\; \omega_{n}}} \right)} = {{- \frac{1}{A}}{\int_{0}^{\beta}\ {d\; \tau \; e^{i\; \omega_{n}\tau}{\langle{T\; {\rho_{q}(\tau)}\; {\rho_{- q}(0)}}\rangle}}}}} & (26)\end{matrix}$

determines several important quantities such as effectiveelectron-electron interaction, plasmon and phonon spectra, and Friedeloscillations.

$\omega_{n} = \frac{2\; \pi \; n}{\beta}$

are Matsubara frequencies, ρ_(q) is the density operator in q-space andA denotes the area. This quantity is calculated in the canonicalensemble for both of the sub-lattice density operators (ρ=ρ_(a)+ρ_(b)).Eqs. (26)-(33) have been used to derive the dynamical polarization. Thedynamical polarization up to the first order electron-electroninteraction in the long wavelength limit is

$\begin{matrix}{{P^{(1)}\left( {q,{i\; \omega_{n}}} \right)} = {\frac{g_{s}g_{v}}{4\; {\pi \;}^{2}}{\int{d^{2}k\; \Sigma_{s,{\overset{`}{s} = \pm}}{f^{s\overset{`}{s}}\left( {k,q} \right)}\frac{{n_{F}\left( {E^{s}(k)} \right)} - {n_{F}\left( {E^{\overset{`}{s}}\left( {{k + q}} \right)} \right)}}{{E^{s}(k)} - {{E^{\overset{`}{s}}\left( {{k + q}} \right)}i\; \hslash \; \omega_{n}}}}}}} & (27)\end{matrix}$

where g_(s)=g_(v)=2 are the spin and valley degeneracy, n_(F) is theFermi distribution and E^(s)(k)=s

v_(F)k−E_(F) is the graphene energy. The band-overlap of wavefunctions,f^(sś)(k,q), is a specific property of graphene

$\begin{matrix}{{f^{s\overset{`}{s}}\left( {k,q} \right)} = {\frac{1}{2}\left( {1 + {s\overset{`}{s}\frac{k + {q\; \cos \; \phi}}{{k + q}}}} \right)}} & (28)\end{matrix}$

where φ signifies the angle between k and q.

Integration over φ and k gives the retarded polarization orcharge-charge correlation function

P ⁽¹⁾(q,ω)=P ₀ ⁽¹⁾(q,ω)+ΔP ⁽¹⁾(q,ω)  (29)

where

$\begin{matrix}{{P_{0}^{(1)}\left( {q,\omega} \right)} = {{- i}\; \pi \frac{\left( {q,\omega} \right)}{\hslash^{2}v_{F}^{2}}\mspace{14mu} {and}}} & (30) \\{{\Delta \; {P^{(1)}\left( {q,\omega} \right)}} = {{- \frac{{gE}_{F}}{2{\pi\hslash}^{2}v_{F}^{2}}} + {\frac{\left( {q,\omega} \right)}{\hslash^{2}v_{F}^{2}}\left\{ {{\left( \frac{{\hslash\omega} + {2E_{F}}}{\hslash \; v_{F}q} \right)} - {{\Theta \left( {\frac{{2E_{F}} - {\hslash\omega}}{\hslash \; v_{F}q} - 1} \right)} \times \left\lbrack {{\left( \frac{{2E_{F}} - {\hslash\omega}}{\hslash \; v_{F}q} \right)} - {i\; \pi}} \right\rbrack} - {{\Theta \left( {\frac{{\hslash\omega} - {2E_{F}}}{\hslash \; v_{F}q} + 1} \right)}\left( \frac{{\hslash\omega} - {2E_{F}}}{\hslash \; v_{F}q} \right)}} \right\}}}} & (31)\end{matrix}$

Two functions

(q,ω) and

(x) are defined as

$\begin{matrix}{{\left( {q,\omega} \right)} = {\frac{g}{16\pi}\frac{\hslash \; v_{F}^{2}q^{2}}{\sqrt{\omega^{2} - {v_{F}^{2}q^{2}}}}\mspace{14mu} {and}}} & (32) \\{{(x)} = {{x\sqrt{x^{2} - 1}} - {\ln \left( {x + \sqrt{x^{2} - 1}} \right)}}} & (33)\end{matrix}$

where g=g_(s)g_(v)=4.

For ω>qv_(F) and in the long wavelength limit q→0,

${x = {{\frac{{\hslash\omega} \pm {2E_{F}}}{\hslash \; v_{F}q}}1}},$

so x²−1≈x² and

(x)≈x²−2 ln(x). We derive here the dynamical polarization (Eq. (38)) andthe effective dielectric of graphene on the substrate (Eq. (51)) inthese regimes. The expansion of

(q, ω) gives

$\begin{matrix}{{\left( {q,\omega} \right)} = {{\frac{g}{16\pi}\frac{\hslash \; v_{F}^{2}q^{2}}{\omega}\left( {1 - \frac{v_{F}^{2}q^{2}}{\omega^{2}}} \right)^{{- 1}/2}} \approx {\frac{g}{16\pi}\frac{\hslash \; v_{F}^{2}q^{2}}{\omega}\left( {1 + \frac{v_{F}^{2}q^{2}}{2\omega^{2}}} \right)}}} & (34)\end{matrix}$

In this condition and for intraband transition (ℏω<2μ)

$\begin{matrix}{{{\left( \frac{{\hslash\omega} + {2E_{F}}}{\hslash \; v_{F}g} \right)} - {\left( \frac{{2E_{F}} - {\hslash \; \omega}}{\hslash \; v_{F}q} \right)}} = {\frac{8{\hslash\omega}\; E_{F}}{\hslash^{2}v_{F}^{2}q^{2}} + {2\mspace{11mu} {\ln \left( {\frac{{2E_{F}} - {\hslash\omega}}{{2E_{F}} + {\hslash\omega}}} \right)}}}} & (35)\end{matrix}$

As a result, ΔP⁽¹⁾(q,ω) reduces to

$\begin{matrix}{{\Delta \; {P^{(1)}\left( {q,\omega} \right)}} = {{{- \frac{{gE}_{F}}{2{\pi\hslash}^{2}v_{F}^{2}}} + {\frac{\left( {q,\omega} \right)}{\hslash^{2}v_{F}^{2}}\left\{ {\frac{8{\hslash\omega}\; E_{F}}{\hslash^{2}v_{F}^{2}q^{2}} + {2\mspace{14mu} {\ln \left( {\frac{{2E_{F}} - {\hslash\omega}}{{2E_{F}} + {\hslash\omega}}} \right)}} + {i\; \pi}} \right\}}} = {\frac{{gq}^{2}}{8{\pi\hslash\omega}}\left\{ {\frac{2E_{F}}{\hslash \; \omega} + {\frac{1}{2}{\ln \left( {\frac{{2E_{F}} - {\hslash\omega}}{{2E_{F}} + {\hslash\omega}}} \right)}} + \frac{i\; \pi}{2}} \right\}}}} & (36)\end{matrix}$

If 2E_(F)>>ℏω

$\begin{matrix}{{\Delta \; {P^{(1)}\left( {q,\omega} \right)}} = {\frac{g\; q^{2}}{8{\pi\hslash}\; \omega}\left\lbrack {\frac{2E_{F}}{\hslash \; \omega} + \frac{i\; \pi}{2}} \right\rbrack}} & (37)\end{matrix}$

By taking the decay rate ω→ω+iτ⁻¹ into account and substituting Eq. (30)into Eq. (29), the dynamical polarization reduces to

$\begin{matrix}{{P^{(1)}\left( {q,\omega} \right)} \approx \frac{E_{F}q^{2}}{{{\pi\hslash}^{2}\left( {\omega + {i\; \tau^{- 1}}} \right)}^{2}}} & (38)\end{matrix}$

The electron life time (τ) can be derived by considering the impurity,electron-phonon interaction and the scattering related to nanostructureedges

τ=τ_(DC) ⁻¹+τ_(edge) ⁻¹+τ_(e-ph) ⁻¹  (39)

where τ_(DC)=95 fs is the lifetime measured from Drude response of thepristine graphene. It can be evaluated via the measured DC mobility (μ)of the graphene sample through

$\begin{matrix}{\tau_{DC} = {\frac{\mu\hslash}{e\; V_{F}}\sqrt{\pi \; n}}} & (40)\end{matrix}$

where V_(F)˜10⁶ m/s is the Fermi velocity and n=(E_(F)/

V_(F))²/π is the charge carrier density.

$\tau_{edge} \approx \frac{3 \times 10^{6}}{w - w_{0}}$

is due to the scattering from the nanostructure edges, and τ_(e-ph)=

/2

(Σ_(e-ph)) is related to the scattering because of coupling of electronsand phonons

(Σ_(e-ph))=γ|

ω−sgn(

ω−E _(F))

ω_(op)|  (41)

where Σ_(e-ph) is the electron self-energy, γ=18.3×10⁻³ is adimensionless constant describing the electron-phonon couplingcoefficient, and

ω_(oph)≈0.2 eV is the graphene optical phonon energy.

In the presence of the optical phonons, the effective dielectricfunction can be calculated via RPA expansion of the dielectric function

ε^(RPA)(q,ω)=ε_(m) −v _(c)(q)P ⁽¹⁾(q,ω)−ε_(m)Σ_(l) v _(sp,l)(q,ω)P⁽¹⁾(q,ω)−ε_(m) v _(op)(q,ω)P _(j,j) ¹(q,ω)   (42)

where

$ɛ_{m} = \frac{ɛ_{1} + ɛ_{2}}{2}$

is the average of dielectric constants of graphene's environment. Thesecond term represents the effective Coulomb interaction of electrons ingraphene, and

$v_{c} = \frac{e^{2}}{2q\; ɛ_{0}}$

is the direct Coulomb interaction. The third term is the effectivedielectric function for different phonon modes (l) coming fromelectron-electron interaction mediated by substrate optical phonons,which couple to the electrons by means of the Fröhlich interaction, i.e.

v _(sph,l)(q,ω)=|M _(sp)|² G _(l) ⁰(ω)  (43)

where |M_(sph)|² is the scattering matrix element given by

$\begin{matrix}{{M_{sph}}^{2} = {\frac{\pi \mspace{11mu} e^{2}}{ɛ_{0}}\frac{e^{{- 2}{qz}_{0}}}{q}\mathcal{F}^{2}}} & (44)\end{matrix}$

where z₀ is the distance between the graphene and the substrate, and

² denotes the Fröhlich coupling strength. The free phonon Green'sfunction G_(l) ⁰ is defined as

$\begin{matrix}{{G_{l}^{0}(\omega)} = \frac{2\omega_{{sp},l}}{\hslash \left( {\left( {\omega + \frac{i\; \hslash}{\tau_{{sp},l}}} \right)^{2} - \omega_{{sp},l}^{2}} \right)}} & (45)\end{matrix}$

where ω_(sph) and τ_(sph) are the substrate phonon frequency andlifetime, respectively. The last term of Eq. (42) corresponds tographene's optical phonon mediated electron-electron interaction

v _(op)(q,φ)=|M _(oph)|² G ⁰(ω)  (46)

Here |M_(oph)|² defines the scattering matrix element

$\begin{matrix}{{M_{oph}}^{2} = \frac{{\hslash g}_{0}^{2}}{2\rho_{m}\omega_{oph}}} & (47)\end{matrix}$

where g₀=7.7 eV/A^(o) is the coupling constant, ρ_(m) is the massdensity of graphene, and ω_(op) is the graphene optical phononfrequency. Similar to the substrate phonon case, G^(o)(ω) is the freephonon Green's function

$\begin{matrix}{{G^{o}(\omega)} = \frac{2\; \omega_{oph}}{\hslash \left( {\left( {\omega + \frac{i\; \hslash}{\tau_{{op}\; h}}} \right)^{2} - \omega_{op}^{2}} \right)}} & (48)\end{matrix}$

where T_(oph) is the graphene optical phonon lifetime. In Eq. (42),P_(j,j) ¹(q, ω) is the current-current correlation function which isrelated to the retarded polarization by means of the charge continuityequation

$\begin{matrix}{{P_{j,j}^{1}\left( {q,\omega} \right)} = {{\frac{\omega^{2}}{q^{2}}{P^{(1)}\left( {q,\omega} \right)}} - {\frac{v_{F}}{q^{2}}{\left\lbrack {{q \cdot {\hat{J}}_{q}},{\hat{\rho}}_{- q}} \right\rbrack\rangle}}}} & (49)\end{matrix}$

where Ĵ_(q) is the single-particle current operator in q-space. Sincethe second term is purely real, the imaginary part of P_(j,j) ¹(q, ω)can be calculated by evaluating imaginary part of the first term.

Collective oscillation of electron modes can be obtained by settingε^(RPA)(q,ω)=0. The extinction function is identified as

${Z = {- \frac{\delta \; T}{T_{0}}}},$

or for the plasmonic structure coupled to an optical cavity

${Z = {- \frac{\delta \; R}{R_{0}}}},$

where δR=R−R₀ and R/R₀ is the reflectance with/without plasmonexcitation, which corresponds to the enhanced absorbance at resonancefrequencies

$\begin{matrix}{{Z\text{\textasciitilde}} - {\left( \frac{1}{ɛ^{RPA}} \right)}} & (50)\end{matrix}$

In the long wavelength regime, by substituting Eq. (38) and v_(c) intoEq. (42), the second term on the right hand side is reduced to the Drudemodel dielectric function

$\begin{matrix}{ɛ_{Drude} = {{{- {v_{c}(q)}}{P^{(1)}\left( {q,\omega} \right)}} = {- \frac{e^{2}E_{f}q}{2\; ɛ_{0}{{\pi\hslash}^{2}\left( {\omega + {i\; \tau^{- 1}}} \right)}^{2}}}}} & (51)\end{matrix}$

According to Eq. (25), the in-plane momentum of the pristine grapheneshould be equal to

$q = {\frac{2}{t}.}$

So, the effective dielectric function of graphene on the substrate isgiven by

ε^(RPA)(q,ω)=ε^(Drude)−ε_(m)Σ_(l) v _(sp,l)(q,ω)P ⁽¹⁾(q,ω)−ε_(m) v_(op)(q,ω)P _(j,j) ¹(q,ω)  (52)

In this dielectric function, the phonon terms, which are small relativeto ε^(Drude), perturb the original system. In order to include theelectron-phonon coupling in the simulation and to predict theexperimental results with higher accuracy, Eq. (52) has been used as theinput data in the FDTD simulations to generate the red diagram of FIG.4b in the main manuscript.

Referring now to FIG. 10, a diagram 180 is now described. The diagram180 shows light absorption of cavity coupled patterned graphene withcavity thickness of L=1400 nm, Period=400 nm, Diameter=330 nm, E_(f)=1.0eV and μ=960 cm²/V·s for different auto shutoff mins.

Referring now to FIGS. 11a-11b , diagrams 185, 190 are now described.The diagrams 185, 190 show electrical conductivities of monolayergraphene sheets with different carrier mobilities. Experimental resultsand analytical calculations for FIG. 11a α=250 cm²/V·s and FIG. 11bμ=960 cm²/V·s show the dependence of electrical conductivity on Fermienergy. The dotted line is the electrical conductivity byσ(E_(F))=σ_(min)(1+E_(F) ⁴/Δ⁴)^(1/2) and the solid line demonstratesα=ρeμ.

Referring now to FIGS. 12a-12b , diagrams 195, 200 are now described.The diagrams 195, 200 show, in FIG. 12a , optical absorption ofpristine, patterned and cavity coupled patterned graphene with cavitythickness of L₁, Period-400 nm, Diameter=330 nm, E_(f)=1.0 eV and μ=960cm²/V·s. The first and modes are shown by λ₀ and λ₁, respectively. Theinset images show the intensity of electric field distribution in zdirection derived from FDTD for different plasmonic modes. The whitecircle line shows the graphene edges. In FIG. 12b , the experimentalresult for the light absorption of the whole structure without graphene(square marked line), and the total light absorption of doped patternedgraphene and substrate/superstrate (x marked line: high mobility tocircle marked line: low mobility), pristine graphene andsubstrate/superstrate (solid line) and bare pristine graphene (solidline).

Referring now to FIG. 13, an optical detector device 20 according to thepresent disclosure is now described. The optical detector device 20illustratively includes a substrate 21 (e.g. glass), and a reflectorlayer 22 carried by the substrate. The optical detector device 20illustratively includes a first dielectric layer 23 over the reflectorlayer 22, and a graphene layer 24 over the first dielectric layer andhaving a perforated pattern 25 therein.

The perforated pattern 25 illustratively includes a square array ofopenings 26 a-26 c. For example, in the illustrated embodiment, each ofthe openings 26 a-26 c is circle-shaped. In other embodiments (notshow), the openings 26 a-26 c may have another shape, such as a square,an oval, or a triangle. The perforated pattern 25 is illustrativelysymmetrical about longitudinal and transverse axes. The first dielectriclayer 23 may comprise one or more of an ion gel, a polymer material, anda SU-8 epoxy-based negative photoresist, for example. Also, in theillustrated embodiment, the graphene layer 24 includes a monolayer ofgraphene (i.e. a layer having a thickness of one molecule).

Also, the optical detector device 20 illustratively includes a seconddielectric layer 27 over the graphene layer 24, a first electricallyconductive contact 29 coupled to the second dielectric layer (e.g.polymer material), and a second electrically conductive contact 28coupled to the graphene layer. The second dielectric layer 27 maycomprise one or more of an ion gel, and a polymer material, for example.

In some embodiments, the first and second electrically conductivecontacts 28, 29 each comprises one or more of aluminum, palladium,copper, gold, and silver. The reflector layer 22 may comprise goldmaterial, for example. In some embodiments, the reflector layer 22 maycomprise a gold backed mirror. The reflector layer 22 may have athickness greater than a threshold thickness for optical opacity.

Another aspect is directed to a method for making an optical detectordevice 10. The method includes forming a reflector layer 22 carried by asubstrate 21, forming a first dielectric layer 23 over the reflectorlayer, and forming a graphene layer 24 over the first dielectric layerand having a perforated pattern 25 therein.

Referring now additionally to FIG. 14, another embodiment of the opticaldetector device 120 is now described. In this embodiment of the opticaldetector device 120, those elements already discussed above with respectto FIG. 13 are incremented by 100 and most require no further discussionherein. This embodiment differs from the previous embodiment in thatthis optical detector device 120 illustratively includes an unpatternedgraphene layer 124.

Referring now additionally to FIG. 15, another embodiment of the opticaldetector device 220 is now described. In this embodiment of the opticaldetector device 220, those elements already discussed above with respectto FIG. 13 are incremented by 200 and most require no further discussionherein. This embodiment differs from the previous embodiment in thatthis optical detector device 220 illustratively includes the graphenelayer 224 having a perforated pattern 225 with ten rows of sevenopenings 226 a-226 c.

In the following, some additional exemplary discussion now follows.

Design and Simulation Results

An array of nanoholes on graphene conserves the continuity of graphene,and by coupling this perforated graphene to an optical cavity, we showthat it is possible to achieve constructive interference between theincident and scattered electric fields, giving rise to strongenhancement of the absorption. Consequently, the strong light-matterinteraction amplifies direct light absorption in graphene even inconditions of low carrier mobility, unlike other techniques where highcarrier mobility is required for absorption enhancement. The systemconsists of a dielectric slab of thickness L and refractive indexn_(d)=1.56 sandwiched between patterned graphene and an optically thick(200 nm) gold back reflector, as illustrated in FIG. 1d -right (inset).The patterned graphene is obtained by perforating a square array ofholes with 330 nm diameter and 400 nm period. A simple embossing basednanoimprinting technique was followed to pattern the graphene sheet. Thecavity supports transverse electromagnetic modes when the slab thicknesssatisfies the phase equation L=mλ/4n_(eff), where n_(eff) is theeffective refractive index of the dielectric slab, λ is the incidentelectromagnetic wavelength, and m=[1, 2, 3, . . . ] is the m-th order ofthe optical cavity mode. The n_(eff) value, which includes the effect ofpatterned graphene is calculated by the effective medium approach. Thefinite-difference time domain (FDTD) simulations (with auto shutoff minof 10⁻⁸, simulation time of 5000 fs and meshing of 0.05 nm) reveal thatfor odd/even cavity modes excited with x-polarized light, the incidentand reflected electric fields interfere constructively/destructivelygiving rise to a maximum/minimum value in the surface plasmon enhancedabsorption for graphene with electron mobility of μ=960 cm²/V·s andFermi energy of E_(F)=1.0 eV (FIG. 1d ). In the case of destructiveinterference, the incident and reflected electric fields have a phasedifference of n such that their interference results in zero netamplitude. The FDTD absorption spectrum (FIG. 1c ) shows two distinctpeaks at ω₀ and ω₁, which can be attributed to localized surface plasmon(LSP) and propagating surface plasmon (SPP) modes, respectively. This isevident from the corresponding real [Re(E_(z))] part and intensity(|E_(z)|²) of the z-component of the electric field distribution forboth plasmonic modes, as shown in FIG. 12a (inset). The nature of theplasmonic mode at ω₀ is further confirmed to be a LSP because of theclose correspondence between the FDTD and coupled dipole approximation(CDA) modelled absorption spectra of the patterned graphene withoutoptical cavity (FIG. 1c ). Due to the symmetrical square lattice patternof the holes, the excitation of LSP is independent of light polarizationfor normal angle of incidence. The solid white and green dotted lines inthe FDTD calculation in FIG. 1d show the analytical dispersions of thecavity modes as a function of wavelength and cavity thickness, whichaccurately depicts the origin of this extraordinary absorption arisingfrom the temporal and spatial overlap between the LSP resonance and thecavity modes.

The FDTD simulation shows that a cavity length of L=1.6 μm, whichsatisfies the cavity resonance condition, needs to be chosen in order toachieve ˜60% light absorption in patterned graphene at around λ=10 μm,giving rise to about a 30-fold absorption enhancement compared topristine graphene. We use the optical cavity to strongly increase theabsorption of the incident light by means of the enhancement of theelectric field on the patterned graphene. The bare pattern grapheneabsorbs ˜12% of the incident light (FIG. 1c ) which is theoretically andexperimentally enhanced to ˜60% and ˜45% for specific cavity lengths atλ=10 μm, respectively. A comparison between the uncoupled and thecavity-coupled systems (FIGS. 1c and 1d ) shows an increase inabsorption from 12% to 60%, without change in the LSP resonancefrequency for all cavity modes. The FDTD simulation time was set to 5000fs, the “auto shut-off time”, which defines the convergence as 10⁻⁸(this is very small compared to typical simulations for 3Dnanostructures (10⁻⁵)).

The monolayer graphene sheet in FDTD simulation is considered as a bulkmaterial with thickness of 0.5 nm. This means the simulation alwayscompletely converges. Moreover, the periodic boundary condition ensuresbetter convergence. To show the effect of “auto shut-off time” on theresults, the absorption of patterned graphene for different “autoshut-off times” are overlaid in FIG. 10. For all these plots the ripplesare present, which means that the ripples are not artifacts of the FDTDsimulation.

The simulation for shorter time steps and the results were same. Theseripples are the different modes emerging at lower wavelengths because ofdiffraction of surface EM waves. There is no diffraction for theincident light because the period of the pattern is less than thewavelength of the incident light. But, the wavelength of the propagatingsurface wave is much less than that of free space, resulting indiffractions that are seen as ripples. For graphene in an asymmetricdielectric medium, the plasmon wavenumber (k_(p)) can be calculated bymeans of

$\begin{matrix}{{{\frac{ɛ_{1}}{q_{z\; 1}} + \frac{ɛ_{2}}{q_{z\; 2}} + {2\; {\sigma^{intra}(\omega)}}} = 0},} & (1)\end{matrix}$

where ε₁ and ε₂ are dielectric functions of adjacent environments,q_(z1,2)=√{square root over (ε_(1,2)−(k_(p)/k)²)} and k is thewavenumber of incident EM wave. The plasmon diffraction orderscorrespond to the solutions of Eq. (1), which leads to different peaksat lower wavelengths.

For analytical calculation of the optical extinction of the perforatedgraphene in the long wavelength limit, each element is considered as anelectric dipole in the electrostatic limit with a specificpolarizability α(ω). The polarizability of a generalized ellipsoidalnanoparticle is

$\begin{matrix}{{{\alpha (\omega)} = {ɛ_{0}V\frac{ɛ - ɛ_{m}}{ɛ + {L_{e}\left( {ɛ - ɛ_{m}} \right)}}}},} & (2)\end{matrix}$

where ε and ε_(m) are the dielectric functions of the conductive elementand surrounding medium, respectively. V defines the volume, and theshape factor of the ellipsoid, L_(e), is given by:

$\begin{matrix}{{{L_{e} = {\frac{abc}{2}{\int_{0}^{\infty}\frac{d\; q}{\left( {a^{2} + q} \right)\left\{ {\left( {q + a^{2}} \right)\left( {q + b^{2}} \right)\left( {q + c^{2}} \right)} \right\}^{\frac{1}{2}}}}}},}\ } & (3)\end{matrix}$

where a is the diameter of the ellipsoid along the light polarizationdirection, b and c are the diameters along other two dimensions. For thegraphene disk array, a=b=d, where d is the disk diameter and c=t, wheret is the thickness of graphene. To calculate the light absorption ofperforated graphene, the light reflection/transmission of graphene diskarray is used as light transmission/reflection of graphene hole array.This is an approximation to calculate the optical responsivity ofperforated metal by coupled-dipole approximation (CDA) approach.Derivation of the LSP frequency is possible by calculation of the totalelectric potential in presence of two dimensional nanostructureelements. The total electric potential in space is due to thecombination of the radiation of the graphene nanostructure and theexternal electric field, i.e.

$\begin{matrix}{{\varphi (r)} = {{\varphi^{ext}(r)} - {\frac{i}{\omega}{\int_{0}^{d}{\int_{0}^{2\; \pi}{\frac{d^{2}r^{\prime}\mspace{14mu} {\nabla^{\prime}{\cdot {\sigma \left( {r^{\prime},\omega} \right)}}}{\nabla^{\prime}{\varphi \left( r^{\prime} \right)}}}{{r - r^{\prime}}}.}}}}}} & (4)\end{matrix}$

By considering homogeneous doping of graphene, it can be assumed thatthe conductivity does not depend on position, and outside graphene theconductivity goes to zero. It means that σ(r,ω)=

(r)σ(ω), where

(r)=1/0 for inside/outside graphene. By defining a dimensionlessvariable

${\Re = \frac{r}{d}},$

the electric potential is given by

$\begin{matrix}{{\varphi (r)} = {{\varphi^{ext}(r)} + {{\int_{0}^{1}{\int_{0}^{2\; \pi}\frac{d^{2}\Re^{\prime}\; {\nabla^{\prime}{\cdot {{\Gamma –}\left( \Re^{\prime} \right)}}}{\nabla^{\prime}{\varphi \left( \Re^{\prime} \right)}}}{{\Re - \Re^{\prime}}}}}}}} & (5)\end{matrix}$

where

$\begin{matrix}{= {\frac{e^{2}E_{F}}{{\pi\hslash}^{2}ɛ_{m}d}\frac{1}{\omega \left( {\omega + {i\; \tau^{- 1}}} \right)}}} & (6)\end{matrix}$

Equation. (5) introduces a self-consistent potential that in absence ofexternal potential has real eigenvalues related to the plasmonic modes.The LSP frequency is given by

$\begin{matrix}{\omega_{p} = {{\frac{e}{\hslash}\sqrt{\frac{\mathcal{I}\; E_{F}}{\pi \; ɛ_{m}d}}} - \frac{i}{2\; \tau}}} & (7)\end{matrix}$

where

is the eigenvalue of Eq. (5) and can be derived by solving thiseigensystem or by using the results from the FDTD simulation. Theimaginary part of ω_(p) is responsible for the bandwidth of theabsorption peak. In addition, Eq. (7) can be applied for the graphenenanoribbon by replacing d (diameter) with w (nanoribbon width).

The lattice contribution S describes the near field and far fieldcoupling of the electric dipoles

$\begin{matrix}{S = {\sum_{j \neq i}\left\lbrack {\frac{\left( {1 - {ikr}_{ij}} \right)\left( {{3\cos^{2}\theta_{ij}} - 1} \right)e^{{ikr}_{ij}}}{r_{ij}^{3}} + \frac{k^{2}\sin^{2}\theta_{ij}e^{{ikr}_{ij}}}{r_{ij}}} \right\rbrack}} & (8)\end{matrix}$

where r_(ij) is the distance between electric dipoles i and j, θ_(ij) isthe angle between dipole j and {right arrow over (r_(ij))}, and k=ω/cdefines the wavenumber.

The optical reflection coefficient of the disk array can be calculatedby using the polarizability and the lattice contribution

$\begin{matrix}{{r_{disk} = \frac{{\pm i}}{\alpha^{- 1} - S}},} & (9)\end{matrix}$

where

$\begin{matrix}{\; \frac{2\pi \; k}{A}\left\{ \begin{matrix}{\left( {\cos \; \vartheta} \right)^{- 1},{s - {polarization}}} \\{{\cos \; \vartheta},{p - {polarization}}}\end{matrix} \right.} & (10)\end{matrix}$

and ϑ is the incident angle, which is zero in our study, A is the areaof the unit cell, and positive/negative sign stands for s/ppolarization. The transmission coefficient of the disk array can beobtained through t_(disk)=1+r_(disk).

The absorption enhancement further depends on the electron mobility andFermi energy of graphene, which in turn is affected by the choice ofdielectric material, substrate, and gate bias. It is well known thatgraphene on a polymer substrate has a low carrier mobility (<1000cm²/V·s) because of extra scattering processes. Typical scatteringcenters consist of charge impurities, polymers residues, and couplingcenters between graphene electrons and polar or non-polar opticalphonons of the polymer matrix. To study the impact of the reducedcarrier mobility of patterned graphene on its absorption spectra, weperformed FDTD simulations for two different carrier mobilities (μ) of960 cm²/V·s and 250 cm²/V·s. while maintaining the same E_(F) for thecavity-coupled system. In the FDTD simulations, the real and imaginaryparts of grapheme's refractive index (n,k) were calculated from thecarrier mobility using the random phase approximation (RPA). In RPA, forhigh frequencies the complex graphene conductivity is given by

$\begin{matrix}{{\sigma (\omega)} = {\frac{e^{2}\omega}{i\; \pi \; \hslash^{2}}\left\lbrack {{\int_{- \infty}^{+ \infty}{d\; ɛ\; \frac{ɛ}{\omega \left( {\omega + {i\; \tau^{- 1}}} \right)}\frac{d\; {\rho_{F}(ɛ)}}{d\; ɛ}}} - {\int_{- \infty}^{+ \infty}{d\; ɛ\; \frac{{\rho_{F}\left( {- ɛ} \right)} - {\rho_{F}(ɛ)}}{\left( {\omega + {i\; \delta}} \right)^{2} - {4ɛ^{2}}}}}} \right\rbrack}} & (11)\end{matrix}$

where δ→0 is the infinitesimal parameter that is used to bypass thepoles of the integral. The first and second terms correspond to theintraband electron-photon scattering processes and direct electroninterband transitions, respectively. By performing the first integral,the intraband scattering is found to be similar to the Drudeconductivity at low temperature k_(B)T<<E_(F)

$\begin{matrix}{{{\sigma^{intra}(\omega)} \approx {i\; \frac{e^{2}E_{F}}{\pi \; {\hslash^{2}\left( {\omega + {i\; \tau^{- 1}}} \right)}}}},} & (12)\end{matrix}$

where k_(B) is the Boltzmann constant and T is the temperature. At highEM wave frequencies in the visible domain

ω>>(E_(F),k_(B)T) where E_(F) is the Fermi energy with respect to thecharge neutrality point (CNP) of the Dirac cone, interband transitionsdominate and the light absorbance of graphene is A=πα≈2.3%, which isindependent of wavelength (α≈1/137 is the fine structure constant).However, in the mid-IR frequency range and for high Fermi energy E_(F)>>

ω, graphene's optical response is dominated by intraband transitions andthe conductivity (σ) follows the Drude-Lorentz model, i.e. Eq. (12),where T is the relaxation time determined by impurity scattering(τ_(imp)) and electron-phonon (τ_(el-ph)) interaction time asτ⁻¹=τ_(imp) ⁻¹+τ_(el-ph) ⁻¹. According to the charge conservation law,the relation of the bulk current J_(V) and the surface current J_(S) fora material is given by

∫∫J _(S) ds=∫∫∫J _(V) dV,  (13)

which means the relation of two and three dimensional conductivity isdefined by

$\begin{matrix}{{\sigma_{3D} = \frac{\sigma_{2D}}{t}},} & (14)\end{matrix}$

where t describes the thickness of the material. The dielectric functionof graphene can be obtained via its AC conductivity by means of

$\begin{matrix}{{{ɛ(\omega)} = {ɛ_{g} + \frac{i\; \sigma_{3D}}{ɛ_{0}\omega}}},} & (15)\end{matrix}$

where ε_(g)=2.5 is the dielectric constant of graphite. Substituting Eq.(14) into Eq. (15) gives the in-plane dielectric function of graphene,i.e.

$\begin{matrix}{{{ɛ(\omega)} = {{ɛ_{g} + \frac{i\; \sigma^{intra}}{ɛ_{0}\omega \; t}} = {ɛ_{g} - \frac{e^{2}E_{F}}{{\pi\hslash}^{2}ɛ_{0}{\omega \left( {\omega + {i\; \tau^{- 1}}} \right)}t}}}},} & (16)\end{matrix}$

whereas the surface-normal component is ε_(z)=2.5. The ε(ω) valuescalculated using Eq. (16) were used to obtain the (n,k) values for theFDTD simulations performed for different Fermi energies.

FIG. 1c shows a nominal decrease in the peak absorption from 45% to 31%as the electron mobility is decreased. For a relatively high carriermobility (960 cm²/V·s) loss is small and therefore the bandwidth of theabsorption spectrum is narrow, indicating an increased lifetime ofplasmons, as observed in FIGS. 1c-1d for a cavity thickness of L=1.1 μm(this cavity thickness is chosen to show nearby high frequency weakerresonances). Higher loss in lower carrier mobility graphene gives riseto reduced plasmon lifetime and broadening of absorption spectrum, asshown in FIG. 1d . The results from the FDTD simulations demonstratethat our device architecture can induce considerable absorption for lowmobility graphene, which is a significant improvement over previouslystrategized devices that are functional only for high mobility graphene.

Fabrication and Experimental Results

To experimentally verify the results, the cavity-coupled patternedgraphene device was fabricated based on the schematic presented in FIG.3a (see the method section for fabrication details). Large area CVDgrown graphene was transferred on the substrate, and it was verified tobe a monolayer by performing Raman characterization, as shown in FIG. 3b. FIGS. 3c-3d show the scanning electron microscope (SEM) image of ananoimprinted-patterned graphene showing good uniformity in nanoholediameter across the patterned film. Furthermore, the graphene continuityand nano-pattern formation was confirmed by conductive atomic forcemicroscopy (AFM), which shows the difference in conductivity in theholes of the patterned graphene with respect to the surrounding (FIG. 2d).

We used ion gel as the dielectric layer to electrostatically dopepatterned graphene. The measured capacitance of the ion gel layer isC=2.4 μF/cm² and its absorption in mid-IR spectrum is low. The Fermienergy of graphene is given by E_(F)=

v_(F)(πρ)^(1/2), where v_(F)≅10⁶ m/s is the Fermi velocity and n is theelectron/hole density obtained from ρ=CΔV/e, where ΔV is gate voltagerelative to charge neutral point (CNP). The reported Fermi energies arecalculated based on this relation. To estimate the corresponding Fermienergies experimentally, the conductivity of graphene sheet iscalculated based on σ(E_(F))=σ_(min)(1+E_(F) ⁴/Δ⁴)^(1/2,) where σ_(min)is the minimum conductvity and A is the disorder strength parameter. Asshown in FIGS. 11a-11b , by fitting this conductivity to theexperimental data (red dotted line), σ_(min)=0.289 ms/0.371 ms and Δ=297meV/177 meV are obtained for the diagram shown in FIGS. 11a-11b ,respectively. The relation between conductivity and mobility is σ=ρeμ,where μ is the carrier mobility of graphene. Fitting this equation(green solid line) to the experimental results yields μ=250/960 cm²/V·sfor FIGS. 11a-11b . Positive and negative gate voltages correspond ton-doped and p-doped graphene, with a minimum conductivity occurring atthe charge neutral point (CNP). According to this analysis we find thatthe CVD graphene sheet is p-doped during growth and transfer (˜0.05 eV).

For graphene absorption measurement, we followed a well-known techniqueto experimentally measure the reflection spectra of thin films and 2Dmaterials. In the experimental measurement with FTIR, we took thereflection spectrum of the structure shown in FIGS. 14-15 insupplemental material (with unpatterned graphene) as the reference suchthat the FTIR calibrates the spectrum as R=|r|²=1 in the entirewavelength range. Following this, the reflection spectrum (R) of thestructure with patterned graphene is measured with respect to thereference. Due to the presence of the back mirror, the transmission (T)is zero and hence absorption (A)=1−R−T=1−R. This directly yields theabsorption measurements shown in FIGS. 5a-5b, 4a-4b which closelymatches with the FDTD predicted absorption spectra.

The simulated and measured absorption of the pristine graphene,patterned graphene and cavity coupled-patterned graphene are shown inFIG. 12a and FIG. 12b , respectively. FIGS. 5a-5b shows the FDTDsimulated and experimentally measured electronically tunable absorptionspectra of the cavity-coupled devices for high (960 cm²/V·s) (a) and low(250 cm²/V·s) (b) carrier mobility graphene. The carrier mobility isinfluenced by the degree of oxidation and polymer residues on thegraphene surface. In both cases, E_(F) was varied between 0.7 eV to 1.0eV. The high and low carrier mobility graphene devices exhibit a large˜2 μm and ˜1 μm electrostatic tunability, respectively. The smaller peakin FIG. 5a around 7.6 μm corresponds to polymer residue, which shows theeffect of impurities in grapheme's optical response. An increase in theFermi energy leads to an increase in the electron density of graphene(ρ), which strengthens the electric dipole moment generated by the LSPresonance on the nanopatterned edges and therefore enhances lightabsorption, as shown in FIG. 5a . As seen from FIG. 5a , there is a goodagreement between Coupled Dipole Approximation (CDA) predictions,experimental measurements, and analytical graphene plasmon frequencyω_(p)∝√{square root over (E_(F))}∝ρ^(1/4). According to the experimentalabsorption spectra, the plasmon lifetimes (τ_(PL)=

Γ⁻¹) for high (960 cm²/V·s) and low (250 cm²/V·s) carrier mobilitygraphene are determined to be τ_(PL(high))≈38 fs and τ_(PL(low))≈16 fs,respectively, which is compatible with the momentum relaxation time (τ).

Plasmon-Phonon Coupling

While the theoretical prediction using the FDTD method is in excellentagreement with the LSP peak locations (ω₀) in the experimental curves(FIG. 4b ), it fails to explain the asymmetric line-shape of theresonance. Hence, we can infer that in our device the effectivecombination of SU-8 polymer and the ion-gel matrix behaves as a polarsubstrate. Polar materials have ions of different valence, whoseoscillating dipole moment gives rise to the interaction betweenelectrons and optical phonons-called the Fröhlich interaction. Thesurface optical phonons in polar substrates are Fuchs-Kliewer like. Byplacing graphene on a polar substrate the long range Fröhlichinteraction mediates the interaction between optical phonons and surfaceplasmons in graphene. The interaction between polar substrate/graphenephonons and electrons in graphene modifies substantially the grapheneplasmon dispersion relation. The white dotted line in FIG. 4a representsthe plasma frequency of graphene.

The dynamic polarizability

$\begin{matrix}{{{\chi \left( {q,{i\; \omega_{n}}} \right)} = {{- \frac{1}{A}}{\int_{0}^{\beta}{d\; \tau \; e^{i\; \omega_{n}\tau}{\langle{T\; {\rho_{q}(\tau)}{\rho_{- q}(0)}}\rangle}}}}},} & (17)\end{matrix}$

determines several important quantities, such as effectiveelectron-electron interaction, plasmon and phonon spectra, and Friedeloscillations.

$\omega_{n} = \frac{2\pi}{\beta}$

are Matsubara frequencies, T is time ordering operator, β=1/k_(B)T,where k_(B) is the Boltzmann constant, and n is an integer number. ρ_(q)is the density operator in q-space and A denotes the area of the sample.This quantity is calculated in the canonical ensemble for both of thesub-lattice density operators (ρ=ρ_(a)+ρ_(b)). The dynamicpolarizability in the RPA regime is given by

$\begin{matrix}{{{\chi^{RPA}\left( {q,\omega} \right)} = \frac{\chi^{0}\left( {q,\omega} \right)}{ɛ^{RPA}\left( {q,\omega} \right)}},} & (18)\end{matrix}$

where χ⁰(q,ω) is the non-interacting (zeroth order) polarizability(single pair bubble) and ε^(RPA)(q,ω)=ε_(m)−v_(c)(q)χ⁰ (q,ω), with ε_(m)being the permittivity of the environment and v_(c)(q)=e²/2ε₀q theCoulomb interaction between the carriers. The RPA method corresponds tothe expansion of 1/ε^(RPA)(q,ω), leading to an infinite power seriesover the bubble diagrams. If optical phonons are also considered, theeffective dielectric function in the RPA expansion takes the form

$\begin{matrix}{{ɛ^{RPA}\left( {q,\omega} \right)} = {ɛ_{m} - {{v_{c}(q)}{\chi^{0}\left( {q,\omega} \right)}} - {ɛ_{m}{\sum\limits_{l}{{v_{{sph},l}\left( {q,\omega} \right)}{\chi^{0}\left( {q,\omega} \right)}}}} - {ɛ_{m}{v_{oph}\left( {q,\omega} \right)}{{\chi_{j,j}^{0}\left( {q,\omega} \right)}.}}}} & (19)\end{matrix}$

The third term is the effective dielectric function for different phononmodes (l) coming from the electron-electron interaction mediated bysubstrate optical phonons, which couple to the electrons by means of theFröhlich interaction, v_(sph,l)(q, ω)=|M_(sph)|²G_(l) ⁰(ω), where|M_(sph)|² is the scattering and G_(l) ⁰ is the free phonon Green'sfunction. The last term of Eq. (19) corresponds to graphene's opticalphonon mediated electron-electron interaction, v_(oph)(q,ω)=|M_(oph)|²G⁰(ω). Here |M_(oph)|² defines the scattering matrix element and G⁰ (ω)is the free phonon Green's function. In Eq. (19), χ_(j,j) ⁰(q,ω) is thecurrent-current correlation function. By taking the decay rate ω→ω+iτ⁻¹into account, the dynamic polarizability reduces to χ⁰(q,ω)≈E_(F)q²/π

²(ω+iτ⁻¹)². The momentum relaxation time (τ) can be derived byconsidering the impurity, electron-phonon interaction and the scatteringrelated to nanostructure edges τ=τ_(DC) ⁻¹+τ_(edge) ⁻¹+τ_(e-ph) ⁻¹,which determines the plasmon lifetime and the absorption spectrumbandwidth. It can be evaluated via the measured DC mobility μ of thegraphene sample using τ_(DC)=μ

√{square root over (πρ)}/ev_(F), where v_(F)˜10⁶ m/s is the Fermivelocity and Σ=(E_(F)/

v_(F))²/π is the charge carrier density. τ_(edge)≈(1×10⁶/w−w₀)⁻¹ is dueto the scattering from the nanostructure edges, where w is theedge-to-edge distance of the holes, w₀≈7 nm is the parameter thatincludes edge effects, and τ_(e-ph)=

/2Im(Σ_(e-ph)) is related to the scattering because of electron-phononcoupling. Im(Σ_(e-ph))=γ|

ω−sgn(

ω−E_(F))

ω_(oph)|, where Σ_(e-ph) is the electron self-energy, γ=18.3×10⁻³ is adimensionless constant describing the electron-phonon couplingcoefficient, and

ω_(oph)0.2 eV is the graphene optical phonon energy. From this it isevident that the plasmon lifetime is reduced due to the electron-phononinteraction and edge scattering, but the DC conductivity which is usedto calculate the dielectric function of graphene is invariant if theedge-to-edge distance of the pattern is more than the carrier mean freepath (L_(MFP)=v_(F)τ_(DC)). The modified Drude model is not valid for apatterned graphene sheet only if the edge-to-edge distance is muchsmaller than the carrier mean free path of electrons and holes. For thechosen pattern and carrier mobility (μ=960 cm²/V·s), the carrier meanfree path (L_(MFP)=v_(FτDC)<42 nm) is smaller than the edge-to-edgedistance (=70 nm), which means that the modified Drude model is a goodapproximation for the dielectric function of this patterned graphenesheet. In presence of hard boundaries, atomic displacement vanishes atthe boundaries, thereby modifying the acoustic and optical phonondispersion. This means we need to consider a graphene nanoribbon (GNR)with zigzag-edge or armchair-edge and N periods (N is the number atomsbetween two edges) with several quantized vibration modes. This model isapplied in the long wavelength limit; therefore only the lowestvibration modes up to N/2 appear. By applying the boundary conditions tothe displacement equation, the longitudinal (LO) and transverse (TO)optical phonon branches are changed, i.e. ω_(n) ²=ω_(LO) ²−λ²(q_(n)²+q²)²+β_(L) ²(q_(n) ²+q²) and ω_(n) ²=ω_(TO) ²−β_(T) ²(q_(n) ²+q²).This means the optical phonon frequency, which is almost the same forboth branches (LO and TO), shifts from ω_(op)˜1581 cm⁻¹ to ω_(op)˜1591cm⁻¹ for both zigzag-edge and armchair-edge GNR. We used this modifiedoptical phonon frequency in FIGS. 4a-4b . The effect of this change isvery small.

The coupling of plasmon and substrate/graphene phonon can becharacterized through the loss function (Z), which is the imaginary partof inverse effective dielectric function calculated via the generalizedRPA theory

$\begin{matrix}{Z \propto {- {{{Im}\left( \frac{1}{ɛ^{RPA}} \right)}.}}} & (20)\end{matrix}$

The loss function represents the amount of energy dissipated by excitingthe plasmon coupled to the substrate and optical phonons in graphene.The surface plasmons in graphene are damped through radiative andnonradiative processes. Nonradiative damping transfers the plasmonenergy to hot electron-hole excitation by means of intraband transition.FIG. 4a shows the loss function for graphene with carrier mobility μ=960cm2/V·s and E_(F)=1.0 eV. The thickness of the optical cavity is chosento be 1.1 μm such that the first (ω₀) and second (ω₂) modes lead to 44%and 33% light absorption, respectively. The plasmon assistedelectron-hole pair generation in this structure lies outside the Landauintraband damping region, indicated by the shaded area in FIG. 4a . Aband gap in the plasmon-phonon dispersion relation is formed viaFröhlich interaction between graphene plasmons and optical phonons. Thiscoupling leads to the splitting of the energy into two distinctbranches: surface plasmon phonon polaritons (SPPPs) and grapheneplasmons (GPs). The horizontal branch line marked as ω₀ is the LSP modein FIGS. 1a and 1s independent of the plasmon wavevector due to thelocalization of the LSP. The asymmetric line shape of the first band(ω₀) in FIG. 1c , which is observed in experiments, is due to themerging of these two bands (LSP and SPPP). FIG. 4a shows a clear blueshift in the GP band at a wavevector (k_(sp)≈5.5×10⁵ cm⁻¹),corresponding to the edge-to-edge distance between the holes in presenceof edge effect. Interestingly, there exists a discrepancy in thelocation of the second mode peak of the FDTD curve simulated withoutaccounting for optical phonons (ω₁) from that of the experimentalspectrum (ω₂). This is attributed to the plasmon-phonon coupling, and weshow that by inserting the plasmon-phonon interaction as a perturbationand using ε^(RPA)(q,ω) in Eq. (19) as effective graphene dielectricfunction in the FDTD simulations, one can recover the experimentallyobserved blue shift, as illustrated in FIG. 4b by the green dotted line.The simple Drude model cannot capture the plasmon-phonon interactionswhich leads to discrepancies between FDTD predictions and experimentalmeasurements as can be observed in FIG. 4b . In the long wavelengthregime, by substituting χ⁰ (q, ω)≈E_(F)q²/π

² (ω+iτ⁻¹)² and v_(c) into Eq. (19), the second term on the right-handside is reduced to the Drude model dielectric function

$ɛ_{Drude} = {{{- {v_{c}(q)}}{\chi^{0}\left( {q,\omega} \right)}} = {- {\frac{e^{2}E_{f}q}{2ɛ_{0}\pi \; {\hslash^{2}\left( {\omega + {i\; \tau^{- 1}}} \right)}^{2}}.}}}$

(21)

According to Eq. (21), the in-plane momentum of the pristine grapheneshould be equal to

$q = {\frac{2}{t}.}$

In Eq. (19), the phonon terms, which are small relative to ε^(Drude),perturb the original system. In order to include the electron-phononcoupling in the simulation and to predict the experimental results withhigher accuracy, Eq. (19) has been used as the input data in the FDTDsimulations to generate the plasmon-phonon dispersion diagram of FIG. 4bwith much improved correspondence between prediction and experimentalobservation. This analysis explains different processes involved in theexperimental results and the physical optoelectronic phenomena andhighlights the plasmon-phonon interaction leads to the hybridization ofthe plasmon dispersion relation, which gives rise to a blue shift in thepropagating surface plasmon spectrum. However, the main absorption peak(ω₀), which originates from LSP, remains unperturbed due to frequencydomain separation between the phonon and LSP resonances.

Conclusion

In conclusion, we have presented a scheme to increase the light-grapheneinteraction by the direct excitation of plasmons on patterned monolayergraphene coupled to an optical cavity. Our design of a square lattice ofholes on graphene, which is experimentally realized following a simplenanoimprinting technique, not only preserves material continuity forelectronic conductivity, which is essential for optoelectronic devices,but also leads to direct plasmon excitation that is independent of theincident light polarization. Therefore, our design outperforms othernanoribbon based devices whose absorption is polarization-dependent,thereby reducing their performance for unpolarized light. This approachtriggers the direct excitation of cavity-coupled plasmon in CVD grownmonolayer graphene with a cavity thickness of L=1.1 μm and yields anexperimentally observed absorption of ˜45%, which is the highest valuereported so far in the 8-12 μm band. We show that a reduction in carriermobility of graphene decreases the absorption to ˜30%, which isnonetheless higher than previous studies. Furthermore, electronicallycontrolled dynamic tunability (˜2 μm) is successfully demonstrated. Wehave shown experimentally and theoretically that the carrier mobility ofgraphene, which is influenced by the defect density, determines theenhanced absorption bandwidth and line-shape. Further, CVD growngraphene quality, pattern, gating optimizations, and alternativelow-absorbance dielectrics as gating materials are needed in order toreach the theoretical maximum absorption of ˜60% for a cavity thicknessof L=1.6 μm. Such voltage tunable high absorption in monolayer graphenewill enable the development of various practical graphene basedoptoelectronic devices like photodetectors, sensors, modulators, etc.

Method Section: Device Fabrication Process

The graphene sheet is grown on a 25 μm thick copper foil in an ovencomposed of a molten silica tube heated in a split tube furnace. Themolten silica tube and copper foil are loaded inside the furnace,evacuated, back filled with hydrogen, and heated up to 1000° C. whilekeeping a 50 sccm H₂ stream. The subsequent steps include reinstatingthe copper foil at 1000° C. for 30 minutes, inserting 80 sccm of CH₄ for30 minutes. Then the furnace is cooled down to room temperature withoutgas feeding. An optically thick layer of Cr/Au (4 nm/200 nm) isdeposited on a glass substrate as a back reflector using e-beamdeposition. A photoresist (SU-8) layer is spin-coated on the gold backreflector to form an optical cavity, that is cured under UV lamp for 2hours and baked on a hot plate for 1 hour at 95° C. in order to completethe cross-linking process. A thin layer (˜20 nm) of Gold-Palladium(Au—Pd) is sputtered on the dielectric spacer which function as a gateelectrode. A CVD-grown graphene sheet is transferred onto the Au—Pdlayer using a PMMA transfer layer which is subsequently dissolved inAcetone. The square lattice hole pattern is fabricated following asimple large area nanoimprinting technique.

A poly dimethylsiloxane (PDMS) stamp is embossed against a thinphotoresist (SU-8) layer that is spun coated on the graphene layer,followed by reactive ion etcher (RIE) in order to perforate the graphenelayer. Low carrier mobility nanomesh graphene is prepared by rinsing theresidual polymers (PMMA and SU-8) in acetone one time for a few seconds.In contrast, the high carrier mobility sample is prepared by repeatingthis process for more than ten times in order to reduce polymer residuesfrom the perforated graphene. A high capacitance ion gel film withrefractive index of 1.3 is drop-casted on graphene in order to tune itsFermi energy to high values (˜1 eV). Ion gel is a printable gatedielectric polymer made by mixing ionic liquid ([EMIM][TFSI])(Sigma-Aldrich, Inc.) with dry PS-PEO-PS (10-44-10 kg/mol) triblockcopolymer (Polymer Source, Inc.) with ratio 1:0.04 in a dry solvent(dichloromethane) (Sigma-Aldrich, Inc.) and by stirring the mixtureovernight. Then it is left for 48 hours inside high vacuum chamber(pressure<10⁻⁶ torr) in order to evaporate the remaining solvent. Thematerials are dried in high vacuum for 24 hours then transferred to theglovebox for 4 days. The gate is fabricated by depositing Cr/Au (3 nm/40nm) on Si substrate. A copper wire is connected to the gate by applyingsilver paste on the side and back. The resulting substrate is flippedupside down and put on top of the ion gel.

Materials Characterization and Measurement

After RIE and the polymer removal, conductive AFM was used to confirmthe presence of a patterned graphene layer on the substrate. Afterpatterning the graphene on copper foil following the same procedure andparameters used to pattern the graphene sheet on the SU-8 layer,conductive AFM (MultiMode, Atomic Force Microscope, Nanoscope III,Digital Instruments, Santa Barbara, Calif.) is employed to map ofconductivity of the patterned graphene with nanoscale spatialresolution. Conductive (Au coated) cantilevers with spring constantk=0.06 N/m was used. Measurements are performed in contact mode and afull IV curve was collected at each pixel of the image. The theoreticalsimulations are done by finite-difference time-domain (FDTD) methodusing Lumerical FDTD (Lumerical Inc.) software. The Raman spectrum ofthe grown graphene sheet is measured by WITec Renishaw RM 1000BMicro-Raman Spectrometer with an excitation laser wavelength of 514 nmand a 50× objective lens. The real and imaginary parts of the golddielectric function used in simulations are taken from Palik. Thecorresponding optical absorption measurements are performed with amicroscope-coupled FTIR (Bruker Inc., Hyperion 1000-Vertex 80). Themobility is measured by using the model 2450 SourceMeter® SMU instrumentand a four-point probe. We applied the gate voltage between bottom andtop gate with ion gel as dielectric in presence of “patterned graphene”with two probes and measured the electrical conductivity throughsource-drain using other probes.

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Many modifications and other embodiments of the present disclosure willcome to the mind of one skilled in the art having the benefit of theteachings presented in the foregoing descriptions and the associateddrawings. Therefore, it is understood that the present disclosure is notto be limited to the specific embodiments disclosed, and thatmodifications and embodiments are intended to be included within thescope of the appended claims.

That which is claimed is:
 1. A method for making an optical detectordevice comprising: forming a reflector layer carried by a substrate;forming a first dielectric layer over the reflector layer; and forming agraphene layer over the first dielectric layer and having a perforatedpattern therein.
 2. The method of claim 1 wherein the perforated patterncomprises a square array of openings.
 3. The method of claim 2 whereineach of the square array of openings is circle-shaped.
 4. The method ofclaim 1 wherein the perforated pattern is symmetrical.
 5. The method ofclaim 1 wherein the first dielectric layer comprises a polymer material.6. The method of claim 1 wherein said graphene layer comprises amonolayer of graphene.
 7. The method of claim 1 further comprising: asecond dielectric layer over said graphene layer; a first electricallyconductive contact coupled to said second dielectric layer; and a secondelectrically conductive contact coupled to said graphene layer.
 8. Themethod of claim 1 wherein said reflector layer comprises gold material.9. The method of claim 1 wherein said reflector layer has a thicknessgreater than a threshold thickness for optical opacity.
 10. A method ofoperating an optical detector device comprising a substrate, a reflectorlayer carried by the substrate, a first dielectric layer over thereflector layer, and a graphene layer over the first dielectric layerand having a perforated pattern therein, the method comprising:performing a direct absorption based upon a cavity coupling with thegraphene layer with the perforated pattern.
 11. The method of claim 10wherein the perforated pattern comprises a square array of openings. 12.The method of claim 11 wherein each of the square array of openings iscircle-shaped.
 13. The method of claim 10 wherein the perforated patternis symmetrical.
 14. The method of claim 10 wherein the first dielectriclayer comprises a polymer material.
 15. The method of claim 10 whereinthe graphene layer comprises a monolayer of graphene.
 16. The method ofclaim 10 further comprising: a second dielectric layer over the graphenelayer; a first electrically conductive contact coupled to the seconddielectric layer; and a second electrically conductive contact coupledto the graphene layer.
 17. The method of claim 10 wherein the reflectorlayer comprises gold material.
 18. The method of claim 10 wherein thereflector layer has a thickness greater than a threshold thickness foroptical opacity.